MAGIC SET THEORY LECTURE NOTES (SPRING
2014)
DAVID ASPERÓ
Contents
1. Introduction
2
2. The axiomatic method: A crash course in first order logic
5
3. Axiomatic set theory: ZFC
9
3.1. The axioms
10
3.2. ZFC vs PA
15
3.3. The consistency question
17
4. Ordinals
20
5. Cardinals
26
6. Foundation, recursion and induction. The cumulative
hierarchy
34
7. Inner models and relativization
38
7.1. Our first relative consistency proof: Con(ZF \{Foundation})
implies Con(ZF)
41
8. Elementary substructures and Skolem closures
42
8.1. Reflection
46
8.2.
–systems
47
9. Forcing
49
9.1. The method of forcing: Partial orders and genericity (an
example)
51
9.2. Formal development of forcing
52
9.3. The forcing relation P
55
10. Two forcing constructions
59
10.1. Adding many Cohen reals
59
10.2. Chain condition and cardinal preservation
60
10.3.
–closure and not adding new reals
62
11. Improving ZFC?
64
References
66
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D. ASPERÓ
1. Introduction
Set theory plays a dual role. It provides a foundation for mathematics and it is itself a branch of mathematics with applications to other
areas of mathematics.
Reducing everything to sets: Set theory was developed / discovered / instigated by Georg Cantor in the second half of the 19th
century, as a result of his investigations of trigonometric series rather
than out of foundational considerations. However, set theory would
soon become the prevalent foundation of mathematics. In fact, it was
born at a time when mathematicians saw the need to define things carefully (i.e., define the object of their study in a mathematical language
referring to reasonably ‘simple’ and well–understood entities) and set
theory provided the means to do exactly that.
Example: What is a di↵erentiable function? What is a continuous
function? What is a function?
A case example: A relation is a set of ordered pairs (a, b). And a
function f is a functional relation (i.e., (a, b), (a, b0 ) 2 f implies b = b0 ).
What is an ordered pair (a, b)? Well, given a, b, we can define
(a, b) = {{a}, {a, b}}
(this definition is due to Kuratowski).
Fact 1.1. Given any ordered pairs (a, b), (a0 , b0 ), (a, b) = (a0 , b0 ) if and
only if a = a0 and b = b0 .
[Easy exercise: Check]
Similarly, for given n, we can define the n–tuple (a0 , . . . , an , an+1 ) =
((a0 , . . . , an ), an+1 ).
So we can successfully define the notion of function from the notion
of set (and the membership relation 2, of course). And the notion of
set is presumably easier to grasp than the notion of function.
What about natural numbers, integers, rational, reals and so on?
We can define 0 = ; (the empty set, the unique set with no elements).
The set ; has 0 members.
We can define 1 = {0} = {;}. The set {;} has 1 member.
We can define 2 = {0, 1} = {;, {;}}. The set {;, {;}} has 2 members.
In general, we can define n + 1 = n [ {n}. With this definition n is a
set with exactly n many members, namely all natural numbers m such
that m < n.
With this definition, each n is an ordinal which is either ; or of the
form ↵ [ {↵} for some ordinal ↵ and all of whose members are either
MAGIC Set Theory lecture notes (Spring 2014)
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the empty set or of the form ↵ [ {↵} for some ordinal ↵, and every ordinal which is either ; or of the form ↵ [ {↵} and all of whose members
are either the empty set or of the form ↵ [ {↵} for some ordinal ↵ is
a natural number (the notion of ordinal, which we will see later on, is
defined only in terms of sets).
What is nice about this is that it gives a definition of the set N of
natural numbers involving only the notion of set:
N is the set of all those ordinals ↵ such that ↵ is either the empty
set or of the form [ { } for some ordinal and such that each of
its members is either the empty set or of the form [ { } for some
ordinal .
+ and · on N can be defined also in a satisfactory way using the
notion of set. Then we can define Z in the usual way as the set of
equivalence classes of the equivalence relation ⇠ on N ⇥ N defined by
(a, b) ⇠ (a0 , b0 ) if and only if a + b0 = a0 + b, and we can define also Q
and the corresponding operations from Z in the usual way.
We can define R as the set of equivalence classes of the equivalence
relation ⇠ on the set of Cauchy sequences f : N ! Q where f ⇠ g if
and only if limn!1 h = 0, where h(n) = f (n) g(n). And so on.
All these constructions involve only notions previously defined together with the notion of set and the membership relation. So they
ultimately involve only the notion of set and the membership relation.
If there is nothing fishy with the notion of set and the operations
we have used to build more complicated sets out of simpler ones, then
there cannot be anything fishy with these higher level objects.
Similarly: We feel confident with the existence of C (which, by the
way, contains “imaginary numbers” like i) once we become confident
with the existence of R and know how to build C from R in a very
simple set–theoretic way.
Also: We can derive everything we know about the higher level objects (like, say, the fact that ⇡ is transcendental) from elementary facts
about sets.
And, presumably, we would expect that the combination of elementary facts about sets can ultimately answer every question we are interested in (is e + ⇡ transcendental?, Goldbach’s conjecture, ...).
Some elementary facts about sets: Given sets A, B, we say that
A is of cardinality at most that of B, and write
|A|  |B|,
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D. ASPERÓ
if there is an injective (or one–to–one) function f : A ! B (remember,
a function is a special kind of set!).
We say that that A and B have the same cardinality, and write
|A| = |B|,
if and only if there is a bijection f : A ! B.
We say that A has cardinality strictly less than B, and write
|A| < |B|,
if and only if there is an injective function f : A ! B but there is no
bijection f : A ! B.
Clearly |A|  |B| and |B|  |C| together imply |A|  |C|. Also, it
is true, but not a trivial fact, that |A| = |B| holds if and only if both
|A|  |B| and |B|  |A| hold (Cantor–Bernstein–Schröder theorem, we
will see this later on).
The notion of cardinality captures the notion of “size” of a set. (Example: |5| < |6|).
Notation: Given a set X, P(X) is the set of all sets Y such that
Y ✓ X. (P(X) is the power set of X).
The following theorem arguably marks the beginning of set theory.
Theorem 1.2. (Cantor, December 1873) Given any set X, |X| <
|P(X)|.
Proof. There is clearly an injection f : X ! P(X): f sends x to the
singleton of x, i.e., to {x}.
Now suppose f : X ! P(X) is a function. Let us see that f cannot
be a surjection: Let
Y = {a 2 X : a 2
/ f (a)}
Y 2 P(X). But if a 2 X is such that f (a) = Y , then a 2 Y if and
only if a 2
/ f (a) = Y . This is a logical impossibility, so there is no such
a.
⇤
This theorem immediately yields that not all infinite sets are of the
same size, and in fact there is a whole hierarchy of infinities! (which
was not known at the time):
|N| < |P(N)| < |P(P(N))| = |P 2 (N)| < . . .
n
. . . < |P
|P n+1 (N)|S< . . .
S (N)| <
n
. . . < | n2N P (N)| < |P( n2N P n (N))| < . . .
More elementary facts:
MAGIC Set Theory lecture notes (Spring 2014)
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Let R be the collection of all those sets X such that
X2
/X
R is a collection of objects, and so (we would naturally say that) it
is therefore a set.
R contains many sets. For instance, ; 2 R, 1 2 R, every natural
number is in R, N 2 R, R 2 R, etc.
Question 1.3. Does R belong to R?
Well, R 2 R if and only if R 2
/ R, which is the same kind of
contradiction that we obtained at the end of the proof of Cantor’s
theorem! So R cannot be a set!! (Russell’s paradox)
So, our naive “theory” of sets is inconsistent and maybe it’s not so
good a foundation of mathematics after all...
Is this the end of the story for set theory?
Well, we like to think in terms of objects built out of sets and like
the simplicity of the foundations set theory was intending to provide.
Also, we find the multiplicities of infinities predicted by set theory an
exciting possibility, and there was nothing obviously contradictory in
Cantor’s theorem.
A retreat: A valid move at this point would be to retreat to a more
modest theory T such that
(1) T should express true facts about sets (or should we say plausible, desirable instead of true?),
(2) T enables us to carry out enough constructions so as to build all
usual mathematical objects (real numbers, spaces of functions,
etc.),
(3) T gives us an interesting theory of the infinite (|N| < |P(N)|,
etc.), and such that
(4) we can prove that T is consistent; or, if we cannot prove that,
such that we have good reasons to believe that T is consistent.
First questions:
(1) What is a theory?
(2) Which should be our guiding principles for designing T ?
We answer (1) first.
2. The axiomatic method: A crash course in first order
logic
For us a theory will be a first order theory. A theory T will always
be a theory in a given language L. It will be a set (!) of L–sentences
expressing facts about our intended domain of discourse.
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D. ASPERÓ
Talk of “sets” of L–sentences before we have even defined T (which
might end up being an intended theory of sets)? Well, those sets of
sentences, as well as the sentences, the language L, etc., are objects in
our meta–theory. Presumably they will obey laws expressible in some
meta–meta–theory (perhaps the same laws the same theory T is, in our
understanding, trying to express!).
A language L consists of
• a (possible empty) set of constant symbols c, d, ...
• a (possibly empty) set of functional symbols f , g, ..., together
with their arities (this arity is a natural number; if f is meant
to represent a function f M : M ! M it has arity 1, if it is
meant to express a function f M : M ⇥ M ! M it has arity 2,
etc.)
• a (possibly empty) set of relational symbols R, S, .... together
with their arities (this arity is again a natural number; if R is
meant to represent a subset RM ✓ M , then it has arity 1, if it
is meant to express a binary relation RM ✓ M ⇥ M , then it has
arity 2, etc.)
These are the non-logical symbols and completely determine L.
We also have logical symbols, which are independent from L:
• ^, _, ¬, !, $
(connectives)
• 8, 9
(quantifiers)
• (, ), =
= is sometimes omitted. Also, many of these symbols are not necessary; we could actually do with just ¬, _ and 9.
Finally, we have a sufficiently large supply of variables: V ar =
{v0 , v1 , . . . , vn , . . .}. For most uses it is enough to take the set of variables to have the same size as the natural numbers.
The language of set theory has only one non–logical symbol, namely
a relational symbol 2 of arity 2.
Let us focus on the language of set theory from now on:
(1) Every expression of the form (vi 2 vj ) or (vi = vj ), with vi and
vj variables, is a formula (an atomic formula).
(2) If ' and are formulas, then (¬'), (' _ ), (' ^ _), (' ^ ),
(' ! ), (' $ ) are formulas. Also, if v is a variable, then
(8v') and (9v') are formulas.
(3) Something is a formula if and only if it is an atomic formula or
is obtained from formulas as in (2).
MAGIC Set Theory lecture notes (Spring 2014)
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When referring to a formula, we often omit parentheses to improve
readability (these expressions are not actual official formulas but refer
to them in a clear way).
A sentence is a formula ' without free variables, i.e., such that every
variable v occurring in ' occurs in some subformula of the form 8v
or of the form 9v .
Examples of formulas are the formulas abbreviated as:
8x8y(x = y $ 8z(z 2 x $ z 2 y))
(The axiom of Extensionality)
8x8y9z8w(w 2 z $ (w = x _ w = y))
or, even more abbreviated,
“for all x, y, {x, y} exists”
(Axiom of unordered pairs).
Another example:
9a9b8y(y 2 x $ ((8w(w 2 y $ (w = a _ w = b))) _ (8w(w 2 y $
w = a)))))
(x is in ordered pair)
The first two formulas are sentences. The third one is not.
Satisfaction:
This takes place of course in the meta–theory:
A pair M = (M, R), where M is a set and R ✓ M ⇥ M , is called an
L–structure.
Given an assignment ~a : Var ! M :
• M |= (vi 2 vj )[~a] if and only if (~a(vi ), ~a(vj )) 2 R.
• M |= (vi = vj )[~a] if and only if ~a(vi ) = ~a(vj ).
• M |= (¬')[~a] if and only if M |= '[~a] does not hold.
• M |= ('0 _ '1 )[~a] if and only if M |= '0 [~a] or |='1 [~a]; and
similarly for the other connectives.
• M |= (9v')[~a] if and only if there is some b 2 M such that
M |= '[~a(v/b)], where ~a(v/b) is the assignment ~b such that
~b(vi ) = ~a(vi ) if v 6= vi and ~b(v) = b.
• M |= (8v')[~a] if and only if for every b 2 M , M |= '[~a(v/b)].
We say that M satisfies ' with the assignment ~a if M |= '[~a].
If ' is a sentence, then M |= '[~a] for some assignment ~a if and only
if M |= '[~a] for every assignment ~a. In that case we say that M is a
model of .
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D. ASPERÓ
Given a set T of formulas and a formula ', we write
T |= '
if and only if for every L–structure M = (M, R) and every assignment
~a : Var ! M , IF M |= [~a] for every 2 T , THEN M |= '[~a].
The relation |= aims at capturing the notion of ‘logical consequence’:
' follows logically from T if and only if ' is true in every world in which
T is true. |= is often called the relation of logical consequence. This
framework is mostly due to the logician A. Tarski (1930’s).
Syntactical deduction
Let T be a set of formulas. We will view T as a set of axioms and
deduce theorems from T : A theorem of T will be the final member n
of a deduction
= ( 0, 1, . . . n)
from T , where we say that = ( 0 , 1 , . . . n ) is a deduction from T if
it is a finite sequence of L–formulas and for every i,
• i is either in T , or
• i is a logical axiom of first order logic,1 or
• i is obtained form j and k , for some j, k < i, by the rule of
Modus Ponens “If ' ! and ', then ” (for all L–formulas
', ). In other word, there are j, k < i and an L–formula '
such that j is ' and k is ' ! i .
If ' is a theorem from T , we write
T `'
` is often called the relation of logical derivability.
Theorem 2.1. (Completeness theorem for first order logic) (K. Gödel,
1930’s) |= = `
A theory T is consistent if no contradiction (say, 9x¬(x = x)) can
be derived from it:
T 0 9x¬(x = x)
Otherwise, it is inconsistent. A theory is inconsistent if and only if
it is trivial, in the sense that it proves everything.
By the completeness theorem the following are equivalent:
• T is consistent.
1A
logical axiom is a member of a certain fixed infinite list – which certainly
does not depend on the theory – expressing universal (logical) truths (e.g., ↵ _ ¬↵
for every L-formula ↵, which of course expresses the fact that if something is not
true then its negation is true).
MAGIC Set Theory lecture notes (Spring 2014)
• There is an L–structure M such that M |= T
some world).
9
(T is true in
We will be interested in whether or not T ` for various choices of
theories T and sentences . The following are equivalent again by (the
contrapositive of) the completeness theorem:
• T 0
• There is an L–structure M such that M |= T but M |= ¬ .
3. Axiomatic set theory: ZFC
Z is for Ernst Zermelo, F is for Abraham Fraenkel, C is for the
Axiom of Choice.
The objects of set theory are sets. As in any axiomatic theory, they
are not defined (they are feature–less objects; in the context of the
theory there is nothing to them apart from what the theory says).
ZFC expresses facts about sets expressible in the first order language
of set theory. The same is true for any other first order theory in
the language of set theory, like ZF, ZFC+“There is a supercompact
cardinal” + ZFC+GCH, ZFC+V = L, ZFC+PFA, ...
Most ZFC axioms will be axioms saying that certain “classes” (built
out of given sets) are actual sets (they are objects in the set–theoretic
universe): Axiom 0, The Axiom of unordered pairs, Union set Axiom,
Power set Axiom, Axiom Scheme of Separation, Axiom Scheme of Replacement and Axiom of Infinity will be of this kind. Here, a class
is any collection of objects, where this collection is definable possibly
with parameters. For example the class of all sets. A proper class
will be a class which is not a set.
ZFC will also have an axiom guaranteeing the existence of sets with
a given property, even if these sets are not definable: The Axiom of
Choice. We will also have two “structural” axioms, namely the Axiom
of Extensionality and the Axiom of Foundation.
A classification of the ZFC axioms.
(1) Structural axioms: Axioms of Extensionality, Axiom of Foundation.
(2) Constructive set–existence axioms: Axiom 0, The Axiom
of unordered pairs, Union set Axiom, Power set Axiom, Axiom Scheme of Separation, Axiom Scheme of Replacement and
Axiom of Infinity.
(3) Non–constructive set–existence axiom: Axiom of Choice.
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D. ASPERÓ
3.1. The axioms. The following is the list of the ZFC axioms.
Axiom of Extensionality: Two sets are equal if and only if they
have the same elements:
8x8y(x = y $ 8z(z 2 x $ z 2 y))
In other words: the identity of a set is completely determined by its
members:
The sets
• ;
• {(a, b, c, n) : an + bn = cn , a, b, c, n 2 N, a, b, c
are the same set.
Axiom 0: ; exists.
2, n
3}
9x8y(y 2 x $ y 6= y)
(of course y 6= y abbreviates ¬(y = y)).
Strictly speaking this axiom is not needed: It follows from the other
axioms. It is convenient to postulate it at this point, though.
In the theory given by the Axiom of Extensionality together with
Axiom 0 we can only prove the existence of one set:
;
So this theory is not so interesting yet. The theory T = {Axiom 0,
Axiom of Extensionality} surely is consistent: For any set a,
({a}, ;) |= T
On the other hand, note that ({a, b}, ;) 6|= T if a 6= b.
Axiom of unordered pairs: For any sets x, y there is a set whose
members are exactly x and y; in other words, {x, y} exists.
8x8y9z8w(w 2 z $ (w = x _ w = y))
Of course, if x = y, then {x, y} = {x} [prove this using the Axiom
of Extensionality.]
Recall that we defined the ordered pair (x, y) as the set {{x}, {x, y}}.
The theory given so far gives us already the existence of infinitely
many sets! For example ;, {;}, {{;}}, {{{;}}}, {{{{;}}}}, {;, {;}},
MAGIC Set Theory lecture notes (Spring 2014)
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{;, {;, {;}}}, {{;}, {;, {;}}}, {;, {;, {;, {;}}}}, ... With the definition
of the natural numbers we have adopted these sets are: 0, 1, {1} =
(0, 0), {{1}} = {(0, 0)}, ((0, 0), (0, 0)), 2 = (0, 1), {0, 2}, {1, 2} = (0, 1),
{0, {0, 2}}, ...
All sets whose existence is proved by the theory given so far have at
most two elements (!). Also, this theory proves the existence of (a, b)
for all a, b.
Union set Axiom: For every set x,
[
x = {y : (9w)(w 2 x ^ y 2 w)}
exists:
S
8x9v8y(y 2 v $ (9w)(w 2 x ^ y 2 w))
SS
x is the set consisting of all the members of members of x,
x
is the set of all the members of members of members of x, etc.
S
Notation: Given sets x, y, x [ y = {a : a 2 x _ a 2 y} = {x, y}.
Note: Given sets x, y, x [ y exists (by the Axiom of unordered pairs
and the Union set Axiom).
With the theory given so far we can prove the existence of: {0} [
{1, 2} = {0, 1, 2} = 3, {0, 1, 2} [ {3} = {0, 1, 2, 3} = 4, {0, 1, 2, 3} [
{4} = {0, 1, 2, 3, 4} = 5, ....
So we can prove the existence of every individual natural number!
Similarly, we can prove the existence of every finite set of natural numbers, every ordered pair of natural numbers, every tuple of natural
numbers, every finite set of tuples of natural numbers, ... However, all
particular sets proved to exist by the theory given so far are finite.
Notation: z ✓ x means:
Every member of z is a member of x.
Power set Axiom: For every x there is y whose elements are exactly
those z which are a subset of x:
8x9y8z(z 2 y $ (8w)(w 2 z ! w 2 x))
Notation: For every a, P(a) = {z : z ✓ a}.
The Power set Axiom says that P(a) is a set whenever a is a set.
With the theory T laid down so far we can prove the existence of
P(n) for any particular n 2 N.
For example:
• P(0) = {;} = 1
• P(1) = {;, {;}} = 2
• P(2) = {;, {;}, {{;}}, {;, {;}}} =
6 4
• ...
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D. ASPERÓ
T is consistent: Let (Xn )n2N be defined recursively by
• X0 = {;}
S
• Xn+1 = Xn [ {{a, b} : a, b 2 Xn } [ { a : a 2 Xn } [ {P(a) :
a 2 Xn }
S
Then ( n2N Xn , 2) |= T .
Actually it would be enough to start with ; and take Xn+1 = P(Xn )
at each stage n + 1.
Note: All particular sets proved to exist by T are still finite.
Axiom Scheme of Separation: Given any set X and any first
order property P ,
{y 2 X : P (y)}
exists; in other words: any definable subclass of a set exists as a set.
8x8v0 , . . . , vn 9y8z(z 2 y $ '(x, z, v0 , . . . vn ))
for every L–formula '(x, z, v0 , . . . vn ) such that y does not occur as
bound (i.e., non–free) variable in it, and where x, y, z, v0 , . . . , vn are
distinct variables.
In the theory laid down so far we can prove the existence of n ⇥ m =
{(a, b) : a 2 n, b 2 m}, and much more.
For a formula '(v0 , . . . vn , u, v), ‘'(v0 , . . . vn , u, v) is functional ’ is an
abbreviation of the formula expressing “for all u there is at most one
v such that '(v0 , . . . vn , u, v)”.2
Axiom Scheme of Replacement: Given any set X and any definable (class)–function F , range(F X) is a set:
“For all x, v0 , . . . , vn , if '(v0 , . . . vn , x, u, v) is functional, then there
is y such that for all v, v 2 y if and only if there is some u 2 x such
that '(v0 , . . . vn , x, u, v),”
for every formula '(v0 , . . . vn , x, u, v) such that y does not occur as
bound variable, and where x, y, u, v, v0 , . . . , vn are distinct variables.
Caution: The Axiom schemes of Separation and Replacement are
not axioms but infinite sets of axioms (!). However, it is obviously
possible to write down a computer program which, given a sentence ,
recognises whether or not belongs to either of these schemes.
Given a set X such that a 6= ; for all a 2 X, a choice function
for X is a function f with dom(f ) = X and such that f (a) 2 a for all
2Which
can be expressed in our language since it has =.
MAGIC Set Theory lecture notes (Spring 2014)
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a 2 X.
Axiom of Choice (AC): Every set consisting of nonempty sets
has a choice function.
Exercise: Write down a sentence expressing the Axiom of Choice.
AC is needed in a lot of mathematics. For example, to prove that
every vector space has a basis, that there are sets of reals which are
not Lebesgue measurable, etc. Nevertheless, historically AC has been
seen with suspicion: Finite sets clearly have choice functions,3 but if
X is infinite, where did the choice function for X come from? Also,
AC has “strange consequences”: For example, it is possible to decompose a sphere S into finitely many pieces and rearrange them, without
changing their volumes – in fact by moving them around and rotating
them, and without running into one another –, in such a way that we
obtain two spheres with the same volume as S!4 This result is known
as the Banach–Tarski paradox.
AC has interesting equivalent formulations (module the rest of ZFC).
For example AC is equivalent to “For every two nonempty sets A, B,
either |A|  |B| or |B| < |A|”. AC is also equivalent to “Every product
of compact topological spaces is compact”.
The Axiom of Foundation: If X 6= ; is a set, there is some a 2 X
such that b 2
/ a for every b 2 X.
In other word: Every nonempty sets has some 2–minimal element.
Modulo the other axioms (in particular AC), the following are equivalent:
• Foundation
• There are no x0 , x1 , . . . , xn , xn+1 , . . . such that . . . 2 xn+1 2
xn 2 . . . 2 x1 2 x0 .
The idea behind Foundation is that sets are generated at di↵erent
stages. If a set X is generated at stage ↵, then all members of X have
been generated at some stage before ↵.
Foundation, together with Extensionality, of course, is perhaps the
most fundamental axiom in set theory.
As with AC, one could perhaps also complain: Where did the 2–
minimal element a of X come from? But wait. a was already in X! If
you remove a from X, X is no longer X.
3Try
4The
to see why.
pieces are not Lebesgue measurable, though.
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D. ASPERÓ
In fact, most people like Foundation: It says that the universe is
generated in an orderly fashion. And it provides a very convenient tool
to use in proofs, which we will be using all the time: Induction.
Let (Vn )n2N be defined by recursion as follows.
• V0 = ;
• Vn+1 = P(Vn )
The theory laid down so far, T = Ax0+ Extensionality + Unordered
Pairs + Union + Power Set + Separation + Replacement + AC +
Foundation, is consistent. In fact
[
( Vn , 2) |= T
n
Still, all sets proved by T to exist are finite. In fact,
[
( Vn , 2) |= “Every set is finite”
n
What do we mean by finite? For the moment let us say that a set X
is finite if and only if for every a 2 X, |X \ {a}| < |X|. Correspondingly, let us say the a set is infinite if and only if it is not finite. This
is not the official definition of ‘finite’ but is equivalent to the official
definition. But it makes things easier to deal with the above ‘definition’
(which does not involve the notion of ordinal, which we haven’t defined
yet).
Axiom of Infinity: There is an infinite set.
Definition 3.1. Given a set x,
(the successor of x).
S(x) = x [ {x}
So, S(0) = 1, S(1) = 2, ... S(n) = n + 1.
The Axiom of Infinity is equivalent to:
(9x)(; 2 x ^ (8y)(y 2 x ! S(y) 2 x))
This is also phrased as: There is an inductive set.
Note: Every inductive set is infinite (in our present sense). To see
this, suppose X is inductive, let a 2 X, and let f : X \ {a} ! X be
the function sending every set of the form S n (a) (for n 2 N, n > 0)
to S n 1 (a), and every set x 2 X which is not of the above form to x
itself. It is easily checked that this function f is a bijection.
One could also define “↵ is an ordinal” (which we will do soon).
Then we would define a natural number as an ordinal ↵ such that
(1) ↵ is either ; or of the form S(y) for some ordinal y and
MAGIC Set Theory lecture notes (Spring 2014)
15
(2) for every x 2 ↵, x is either ; or of the form S(y) for some
ordinal y.
The Axiom of Infinity is then equivalent to:
Axiom of Infinity’: The class of all natural numbers is a set.
In other words: There is some x such that for all y, y 2 x if and only
y is a natural number.
Note that Axiom of Infinity’ is a constructive set–existence axiom,
whereas Axiom of Infinity was not, strictly speaking. Axiom of Infinity’
and Axiom of Infinity are equivalent modulo the other axioms.
The Axiom of Infinity completes the list of ZFC axioms.
Notice the big leap when adding Infinity to the list of axioms. ZFC
certainly proves the existence of infinite sets, by design! Before adding
Infinity we had a theory T which ‘surely’ was consistent.5 Now, with
the addition of Infinity, it’s not so obvious that ZFC is consistent... .
3.2. ZFC vs PA. Peano Arithmetic, also known as PA, is the following
first order theory for (N, S, +, ·, 0), where S(n) = n+1 (in the language
of arithmetic, i.e., the language with S, +, ·, 0):
•
•
•
•
•
•
•
8x(S(x) 6= 0)
8x, y, (S(x) = S(y) $ x = y)
8x(x + 0 = x)
8x, y(x + S(y) = S(x + y)
8x(x · 0 = 0)
8x, y, x · S(y) = x · y + x
8ȳ(('(0, ȳ) ^ (8x('(x, ȳ) ! '(S(x), ȳ))) ! 8x'(x, ȳ))
for every first order formula '(x, ȳ) in the language of arithmetic
(First order Induction Axiom Scheme)
First order arithmetical facts can be expressed in this language, like
for example “· is distributive with respect to +”, Fermat’s last theorem,
Goldbach’s conjecture, ...
PA does prove many facts about (N, S, +, ·, 0). But it does not prove
everything!
5Since
(
S
n2N
Vn , 2) |= T .
16
D. ASPERÓ
Theorem 3.2. (Gödel, 1930’s, Incompleteness Theorem (special case))
If PA is consistent then there is a sentence in the language of arithmetic such that
• PA 0 and
• PA 0 ¬
Gödel’s Incompleteness theorem(s), in their general formulation, are
very profound facts that we will look back into in a moment.
The sentence in the Incompleteness Theorem does not express any
fact that mathematicians would have looked into prior to proving the
incompleteness theorem.
is designed for the purpose of the proof
only.
Notation: Given a set X and n 2 N, let
[X]n = {a ✓ X : |a| = |n|}
Consider the following statement HP:
“For all n, k, m there is some N such that for every colouring f : [N]n
into k colours there is some Y ✓ N such that Y has at least m many
members and at least min(Y ) many members and such that all members of [Y ]n have the same colour under f .”
HP can be easily expressed by a sentence, which I will call HP, in
the language of arithmetic.
ZFC proves that (N, S, +, ·, 0) |= HP. On the other hand:
Theorem 3.3. (L. Harrington and J. Paris, 1977): If PA is consistent, then
PA 0 HP
Consider the theory T = (ZFC \{Infinity}) [ {¬Infinity}. It turns
out that T and PA are essentially the same theory: There are e↵ective
translation procedures
' ! (')
between the sentences in the language of set and the sentences in the
language of arithmetic and
!
( )
between the sentences in the language of arithmetic and the sentences
in the language of set theory such that for all ', ,
• T ` ' if and only if PA ` (')
• PA ` if and only if T ` ( )
MAGIC Set Theory lecture notes (Spring 2014)
17
The Harrington–Paris theorem gives an example of a simple “natural” (purely combinatorial) statement talking only about finite sets
which is true if there is an infinite set but need not be true if there are
no infinite sets (!). Other examples have been found since then.
3.3. The consistency question. We pointed out that the theory
TS= ZFC \{Infinity} was ‘surely’ consistent, based on the fact that
( n2N Vn , 2) |= T (assuming, in our metatheory, that P(a) exists for
every a, that N exists, that the recursive
S construction of F = (Vn )n2N is
well–defined class–function, and that range(F ) exists, i.e., assuming
something like ZFC in our metatheory!)
Question 3.4. Can we prove, in T (equivalently, in PA), that T is
consistent? Can we prove, in ZFC, that ZFC is consistent?
The above questions do make sense: Both T and PA have enough
expressive power to make “T is consistent”, “PA is consistent”, etc. expressible in the theory: For example, we can code formulas, proofs, and
other syntactical notions as natural numbers and reduce a statement
like “PA is consistent” to an arithmetical statement (some specific, but
extremely complex, diophantine equation p(x̄) = 0 does not have solutions). It then makes sense to ask whether T proves that p(x̄) = 0
does not have solutions.
Theorem 3.5. (Gödel’s Incompleteness Theorems) Suppose T is a first
order theory such that
• T recursively enumerable (i.e., there is an algorithm deciding,
for any given sentence , whether or not 2 T ),
• T interprets PA and
• T is consistent.
Then:
(1) There is a sentence such that
• T 0 and
• T 0¬
(First Incompleteness Theorem)
(2) T does not prove that T is consistent (T 0 Con(T ))
(Second Incompleteness Theorem)
A theory T as in (1) is said to be incomplete.
Note: Both ZFC and PA are recursively enumerable. Hence, IF they
are consistent, THEN they are incomplete and they cannot prove their
18
D. ASPERÓ
own consistency. It follows that if we adopt, say, ZFC as our metatheory, we won’t be able to prove any statement of the form “ZFC +
is consistent”. What we can do is prove relative consistency statements of the form “If ZFC is consistent, then ZFC + is consistent”
(Con(ZFC) ! Con(ZFC + )).
On the other hand, note that ZFC ` Con(ZFC \{Infinity}) (equivalently,
ZFC ` Con(PA)): Working within ZFC we can build the set
S
n2N Vn and we can prove
(
[
n2N
Vn , 2) |= ZFC \{Infinity}
We express the above fact by saying that ZFC has consistency strength
strictly larger than ZFC \{Infinity}.
In general, T1 has consistency strength at most that of T0 if and only
if we can prove that if T1 is consistent then T0 is consistent. T1 has
consistency strength strictly larger than T0 if and only if we can prove
“if T1 is consistent, then T0 is consistent”, but we cannot prove ”if T0 is
consistent, then T1 is consistent” unless we can prove “T0 inconsistent.”
And, similarly, we define “T0 and T1 are equiconsistent.”
For example, although ZFC does not prove Con(ZFC), if ZFC is
consistent, it proves that ZFC + and ZFC +¬ are equiconsistent for
many interesting choice of (we will hopefully see examples !).
If T1 has consistency strength strictly larger than T0 then T1 is more
“daring” than T0 . There is a whole natural hierarchy of theories ordered
by consistency strength:
• ZFC is equiconsistent with ZF (= ZFC \{AC}) and is strictly
stronger than ZFC \{Infinity}.
• ZFC + “There is an inaccessible cardinal” is strictly stronger
than ZFC.
• ZFC + “There is a weakly compact cardinal” is strictly stronger
than ZFC + “There is an inaccessible cardinal”.
• ZFC + “There is a measurable cardinal” is strictly stronger
than ZFC + “There is a weakly compact cardinal”.
• ZFC + “There is a Woodin cardinal” is strictly stronger than
ZFC + “There is a measurable cardinal”.
• ZFC + “There is a supercompact cardinal” is strictly stronger
than ZFC + “There is a Woodin cardinal”.
• ZFC + “There is a huge cardinal” is strictly stronger than ZFC
+ “There is a supercompact cardinal”.
• ...
MAGIC Set Theory lecture notes (Spring 2014)
19
Later on I will hopefully say a bit more about the above hierarchy
of so–called ‘large cardinal theories (or axioms).’
Let the Axiom Scheme of Comprehension be: For every formula
'(x̄) in the language of set theory,
9x8y(y 2 x $ '(y, x̄))
Frege’s set theory T consists of the Axiom of Extensionality together
with all instances of the Axiom Scheme of Comprehension. (This was
Frege’s bold attempt to reduce all of mathematics to logic).
T is of course inconsistent by Russell’s paradox.
In which way does ZFC (or ZF) neutralise Russell’s paradox? Well,
ZF proves that there is no R such that for every x, x 2 R if and only
if x 2
/ x: If there was such an R, then R 2 R if and only if R 2
/ R.
So R = {x : x 2
/ x} is, in ZFC, a proper class but not a set.
ZFC is not the only theory of sets that people have considered as
a foundation for mathematics and which neutralises Russell’s paradox
(and other related paradoxes). There are also: Type theories, Quine’s
New Foundations (N F ), etc. However, ZFC is the most well–suited
for developing mathematics. Incidentally, it is worth pointing out that
N F is not known to be consistent relative to any natural extension of
ZFC.
We cannot prove that ZFC is consistent. So why should we feel
confident about its consistency?
The first observation is that the question on the consistency of ZFC
is reducible to the question on the consistency of the smaller theory
ZF:
• ZFC is equiconsistent with ZF: Given any (M, R) |= ZF there
is a LM ✓ M such that (LM , R \ LM ⇥ LM ) |= ZFC (Gödel).
OK, why should we trust ZF then? I will give three reasons next.
• All set–existence axioms of ZF assert the set–hood only of “small
classes.” (This is perhaps vague at this point, but in a little
while you’ll get a clearer picture.) Compare with the Axiom
Scheme of Comprehension, which says that EVERY class is a
set!
• All axioms of ZF are “reasonable” assertions about sets: ZF
says that the set–theoretic universe is exactly the “cumulative
hierarchy”, which provides a very appealing picture of the ‘generation of sets from previously generated sets’ (see later). This
is perhaps the best intrinsic justification for ZF and, as a by–
product, speaks in favour of it consistency: The cumulative
20
D. ASPERÓ
hierarchy looks so natural that it should be a “real object”. It
satisfies the axioms of ZF. Therefore, ZF should not be inconsistent.
• History: No inconsistency has ever been detected within ZF.
We will be working in ZFC until further notice.
4. Ordinals
Definition 4.1. A partial order is an ordered pair (X, R) such that
• R ✓ X ⇥ X,
• for every x 2 X, (x, x) 2 R (R is reflexive),
• for all x, y 2 X, (x, y) 2 R and (y, x) 2 R together imply x = y
(R is anti–symmetric), and
• for all x, y, z 2 X, (x, y) 2 R and (y, z) 2 R together imply
(x, z) 2 R (R is transitive).
We say that R is a partial order on X. Also, we often write xRy for
(x, y) 2 R.
(X, R) is a total ordering (or linear order ) if for all x, y 2 R, either
xRy or yRx.
For example, (N, <), (Q, <) and Q ⇥ Q (with the product order) are
partial orders, and (N, <) and (Q, <) are linear but Q ⇥ Q is not.
Definition 4.2. A binary relation R ✓ X ⇥ X is well–founded if and
only if for every nonempty Y ✓ X there is some a 2 Y which is R–
minimal, i.e., such that (b, a) 2
/ R for every b 2 Y , b 6= a.
For example, the Axiom of Foundation says that 2 \ X ⇥ X is well–
founded for every set X).
Definition 4.3. A well–order is a well–founded linear order.
For example, (5, <) and (N, <) are well–orders, but (R, <) is not.
Note: If (L, ) is a well–order and A ✓ L is nonempty, then min(A)
exists.
Notation: Given a partial order (L, ), < is the relation on L given
by y < x if and only if y  x and y 6= x.
We will sometimes abuse language and say that a pair (L, <) is a
well–order if
• < is transitive,
• < is irreflexive (i.e., x < x fails for all x 2 dom(<))
(i.e., < is a strict order ) and
• < is total (i.e., for all x, y, x < y, y < x or x = y) and
• every nonempty subset of dom(<) has an <–minimal member.
MAGIC Set Theory lecture notes (Spring 2014)
21
Given a partial order (L, ) and x 2 L,
pred(L, x) = {y 2 L : y < x}
A ✓ L is an initial segment of (L, ) i↵ for all x < y 2 L, if
y 2 A, then x 2 A. A ✓ L is a proper initial segment of (L, ) if
A = pred((L, ), a) for some a 2 L.
The following fact is immediate.
Fact 4.4. If (L, <) is a well–order and A ✓ L, then (A, < A) is a
well–order.
Given two partial orders (X0 , 0 ), (X1 , 1 ) a bijection f : X0 ! X1
is an order–isomorphism if and only if for all x, y 2 X0 ,
x 0 y if and only if f (x) 1 f (y).
If there is such an order–isomorphism we write
(X0 , 0 ) ⇠
= (X1 , 1 )
Proposition 4.5. Let (L0 , 0 ) and (L1 , 1 ) be two well–orders. Then
exactly one of the following holds.
(1) (L0 , 0 ) ⇠
= (L1 , 1 )
(2) (L0 , 0 ) is order–isomorphic to some proper initial segment of
(L1 , 1 ).
(3) (L1 , 1 ) is order–isomorphic to some proper initial segment of
(L0 , 0 ).
(Trichotomy)
Proof. Let
f = {(u, v) 2 L0 ⇥ L1 : pred((L0 , 0 ), u) ⇠
= pred((L1 , 1 ), v)}
f is a function: Suppose (u, v), (u, v 0 ) 2 f , v 6= v 0 . Wlog v <1 v 0 .
Since pred((L0 , 0 ), u) ⇠
= pred((L1 , 1 ), v) and pred((L0 , 0 ), u) ⇠
=
0
pred((L1 , 1 ), v ), by composing these order–isomorphisms we obtain
an order–isomorphism
g : pred((L1 , 1 ), v) ! pred((L1 , 1 ), v 0 )
Let v̄ <1 v such that g(v̄) = v. The existence of v̄ shows that
A = {z 2 L1 : z <1 g(z)} =
6 ;
Let z ⇤ = min(A). Let z̄ be such that g(z̄) = z ⇤ . Then g(z̄) = z ⇤ <1
g(z ⇤ ) implies z̄ < z ⇤ and therefore z̄ 2
/ A. Hence, z ⇤ = g(z̄) 1 z̄ <1 z ⇤
⇤
⇤
and therefore z <1 z . Contradiction.
22
D. ASPERÓ
Similarly one can show that for all u, u0 , u 0 u0 i↵ f (u) 1 f (u0 ).
In particular, f is injective.
dom(f ) is an initial segment of (L0 , 0 ): Let u 2 dom(f ) and let
0
u <0 u. There is some v 2 L1 such that there is an order–isomophism
0
0
h : pred(L0 , 0 ), u) ! pred((L1 , 1 ), v)
Let v = h(u ). Then h pred((L0 , 0 ), u0 ) is an order–isomorphism
between pred((L0 , 0 ), u0 ) and pred((L1 , 1 ), v 0 ). This shows (u0 , v 0 ) 2
f.
Similarly one shows range(f ) is an initial segment of (L1 , 1 ).
If either dom(f ) = L0 or range(f ) = L1 , then we are done: In the
first case, either range(f ) = L1 and therefore f is an order–isomorphism
between (L0 , 0 ) and (L1 , 1 ) or else min(L1 \range(f )) = v exists and
f is an order–isomorphism between (L0 , 0 ) and pred((L1 , 1 ), v). In
the second case one proceeds similarly.
Suppose towards a contradiction that L0 \ dom(f ) and L1 \ range(f )
are both nonempty. Let u = min(L0 \ dom(f )) and v = min(L1 \
range(f )). Then f is an order–isomorphism between (pred(L0 , 0 ), u)
and (pred(L1 , 1 ), v) and therefore (u, v) 2 f . But then u 2 dom(f )
and v 2 range(f ). A contradiction.
Finally: It is easy to check that (1)–(3) are mutually exclusive. ⇤
Definition 4.6. A set x is transitive i↵ y ✓ x for every y 2 x. In
other words, x is transitive i↵ y 2 x and z 2 y imply z 2 x.
Examples:
• Every natural number is transitive.
• N is transitive.
• P(N) is transitive.
• {1} is not transitive.
Definition 4.7. A set ↵ is an ordinal if and only if ↵ is transitive and
well–ordered under 2. In other words, letting 2 |↵ be the restriction of
2 to ↵ ⇥ ↵, i.e., the relation on ↵ given by x 2 |↵ y i↵ x 2 y, (↵, 2 |↵)
is a well–order.
Fact 4.8. If ↵ is an ordinal and x 2 ↵, then x is an ordinal and
x = pred((↵, 2 |↵), x)
Proof. Let ↵ be an ordinal and x 2 ↵.
x is transitive: Let z 2 y 2 x. Using the transitivity of ↵ twice we
have that z 2 y 2 ↵ and therefore z 2 ↵. Since 2 |↵ is a transitive
relation (as ↵ is an ordinal), x, y and z are in ↵, and both y 2 x and
z 2 y hold, we must have that z 2 x.
MAGIC Set Theory lecture notes (Spring 2014)
23
The proof that x = pred((↵, 2 |↵), x) is then trivial.
Since both x and ↵ are transitive, (2 |↵) \ (x ⇥ x) = 2 |x. To see
this, note that the following are equivalent for all sets y, z:
• z2y2x
• y 2 ↵ and z 2 ↵ and z 2 y 2 x.
But then 2 |x is a well–order on x since it is the restriction of the
well–order 2 |↵ to x.
⇤
Fact 4.9. If ↵ and
are ordinals and f : (↵, 2) ! ( , 2) is an
order–isomorphism, then f is the identity on ↵. In particular, ↵ = .
Proof. Suppose towards a contradiction that there is a minimal ⇠ 2 ↵
such that f (⇠) 6= ⇠. Since f ⇠ is the identity on ⇠,
f (⇠) = {f (⇠ 0 ) : ⇠ 0 2 ⇠} = {⇠ 0 : ⇠ 0 2 ⇠} = ⇠
which is a contradiction, where the first equality holds since the function f : (↵, 2) ! ( , 2) is an isomorphism.
⇤
Corollary 4.10. (Trichotomy for ordinals) Suppose ↵ and
nals. Then exactly one of the following holds.
are ordi-
(1) ↵ =
(2) ↵ 2
(3) 2 ↵
Corollary 4.11. For every ordinal ↵, ↵ 2
/ ↵.
Corollary 4.12. For all ordinals ↵,
↵2 .
, , if ↵ 2
and
2 , then
Corollary 4.13. If A is a nonempty set of ordinals, then A has an
2–minimal element.
Proof. Let ↵ 2 A. If ↵ is not 2–minimal, then A \ ↵ 6= ;. But then
= min(A \ ↵) exists, and then is an 2–minimal member of A. ⇤
Notation: Ord denotes the class of all ordinals.
The previous corollary says that the relation 2 well–orders Ord. So,
if Ord were a set, it would be an ordinal. But then Ord 2 Ord, and we
have seen that ↵ 2
/ ↵ for every ordinal ↵. Hence we have the following.
Theorem 4.14. Ord is not a set. (Burali–Forti Paradox)
On the other hand:
Fact 4.15. Every transitive set of ordinals is an ordinal.
24
D. ASPERÓ
[Exercise.]
Notation: In the context of ordinals, we will often use < to denote
2. For example, if ↵ and are ordinals, ↵ < means ↵ 2 .
The Burali–Forti Paradox indicates that there should be many ordinals. Here is one:
Fact 4.16. ; is an ordinal.
The following fact shows how to generate the least ordinal bigger
than a given ordinal.
Fact 4.17. If ↵ is an ordinal, then S(↵) = ↵ [ {↵} is an ordinal.
Proof. If y 2 S(↵), then either y 2 ↵ or y = ↵. In the first case,
y ✓ ↵ [ S(↵). In the second case, y = ↵ ✓ ↵ [ {↵}. Hence S(↵) is
transitive.
Every member of S(↵) is either a member of ↵ or is ↵, and hence is an
ordinal and therefore transitive. It follows that 2 |S(↵) is a transitive
relation, and it can be shown similarly that it is linear.
Finally, if X ✓ ↵[{↵} is nonempty and X \↵ 6= ;, then a 2–minimal
member of X \ ↵ (which exists since ↵ is an ordinal) is 2–minimal in
X. The other case is when X = {↵}. Then ↵ is 2–minimal.
⇤
Definition 4.18. An ordinal is a successor ordinal if and only if it is
of the form S(x). It is a limit ordinal if and only if it is not a successor
ordinal (so, ; is a limit ordinal).
Definition 4.19. A natural number is an ordinal which is either ;
or a successor ordinal and such that all its members are either ; or a
successor ordinal.
A set is finite if and only if it bijectable with a natural number.
Notation: Given any ordinal ↵, ↵ + 1 = S(↵).
The set of all natural numbers is denoted by !. ! exists by the
Axiom of Infinity.
! is an ordinal since it is a transitive set of ordinals. It is the least
nonzero limit ordinal.
!+1 = S(!) = ![{!}, (!+1)+1 = S(!+1) = (![{!})[{![{!}},
etc. are successor ordinals.
We will automatically view ordinals ↵ as embedded with the relation
2 |↵ well–ordering them.
Lemma 4.20. Let (L, ) be a well–order and ↵ an ordinal. Then there
is at most one order–isomorphism f : (L, ) ! (↵, 2).
MAGIC Set Theory lecture notes (Spring 2014)
25
This lemma is immediate since the composition of order–isomorphisms
is an order–isomorphism, the inverse of an order–isomorphism is an
order–isomorphism, and since the identity is the only order–isomorphism
between (↵, 2) and itself.
Theorem 4.21. Every well–order (L, ) is order–isomorphic to a unique
ordinal.
Proof. By what we have seen it suffices to prove that (L, ) is order–
isomorphic to some ordinal.
Suppose, for a contradiction, that
{y 2 L : pred((L, ), y) 6⇠
6 ;
= (↵, 2) for any ↵ 2 Ord} =
and let
x = min{y 2 L : pred((L, ), y) ⇠
6= (↵, 2) for any ↵ 2 Ord}
By the lemma, for all z < x let ↵z be the unique ordinal such that
(pred((L, ), z), ) ⇠
= (↵z , 2) and let
fz : (pred((L, ), z), ) ! (↵z , 2)
be the corresponding unique order–isomorphism. Then, again by the
lemma, if z < z 0 < x, then ↵z 2 ↵z0 and fz = fz0 pred((L, ), z).
Assume max(pred((L, ), x) does not exist (the proof in the other
case is similar [Exercise]). Let now
f : (pred((L, ), x), ) ! Ord
be given by f (y) = fy0 (x) for any y 0 such that y < y 0 < x. This function
is well–defined by the above and it is easy to see that it is an order–
isomorphism between pred((L, ), x) and (range(f ), 2). But range(f )
is a set, by Replacement, and it is transitive. Therefore it is an ordinal.
This contradicts the choice of x. We thus have that for every x 2 L
there is a unique ordinal ↵x such that there is an order–isomorphism
fx : pred(L, ), x) ! (↵, 2),
and this isomorphism is unique.
Now, arguing as above, we can glue together all these isomorphisms
into an isomorphism f : (L, ) ! (X, 2), where X is a transitive set
of ordinals and therefore an ordinal.
⇤
Given a well–order (L, ), the unique ordinal ↵ such that (L, ) ⇠
=
(↵, 2 |↵) is the order type of (L, ), denoted ot((L, )).
Many sets can be well–ordered in di↵erent ways (so that the corresponding well–orders have di↵erent order types). For example, ! can
be well–ordered by 2 in order type !. And it can be well—ordered by
putting 0 on top of every n > 0 and well-ordering ! \ {0} according to
26
D. ASPERÓ
2. This well–order has order type ! + 1.
[Exercise: Characterize the sets that can be well–ordered in di↵erent
ways.]
5. Cardinals
We have seen the ordinals ! + 1, (! + 1) + 1, ((! + 1) + 1) + 1, etc.,
aka !, ! + 1, ! + 2, etc. The set consisting of all natural numbers and
! + n for every n < ! is a transitive set of ordinals and therefore also
an ordinal. It is called ! + !. We can then build (! + !) + 1, and so
on. All these ordinals are countable (i.e., they are bijectable with !).
Question 5.1. Is there an infinite ordinal which is not bijectible with
!?
Definition 5.2. A cardinal is an ordinal  such that  is not bijectible
with any ordinal ↵ < .
So, each natural number is a cardinal, ! is a cardinal, but no ! + n,
is a cardinal. And the same goes for ! + !, (! + !) + 1, etc.
When regarded as a cardinal, ! is also denoted @0 (so N = ! = @0 ).
Notation: If X is bijectable with a cardinal , we say that  is the
cardinality of X and write |X| = .
We will see that ZFC proves that every set is bijectable with an
ordinal. Caveat: In a context without AC one can extend the notion of
cardinals to things that are not ordinals in a perfectly meaningful way.
We don’t need to do that for the moment. So, for us, at least for the
moment, cardinals are ordinals. Cardinals, in our sense, are sometimes
also called ‘alephs’.
Definition 5.3. !1 , also denoted @1 , is the first uncountable cardinal
(in other words, the first infinite cardinal not bijectable with !).
Proposition 5.4. !1 exists.
Proof. Say that X ✓ ! codes a well–order if
{(n, m) 2 ! ⇥ ! : 2n+1 3m+1 2 X}
is a well–order of !.
Every infinite initial segment of a well–order coded by a subset of !
can be coded by a subset of ! ([Exercise]). Hence = ! [ {↵ : ↵ =
ot((!, )) for some  coded by some X ✓ !} is transitive and is a
set since it is ! [ range(F ), where F : P(!) ! Ord is the function
MAGIC Set Theory lecture notes (Spring 2014)
27
sending X to ot((!, )) if X codes  and to 0 otherwise. Hence is
an ordinal.
is not countable: If f :
! ! were a bijection, {2n+1 3m+1 :
f 1 (n) 2 f 1 (m)} would be a subset of ! coding a well–order of order
type . But then 2 , which is impossible for ordinals.
⇤
It is easy to see that the ordinal
that is,
in the above proof is precisely !1 ;
!1 = ! [ {↵ : ↵ = ot((!, )) for some  coded by some X ✓ !}
[Exercise]
Similarly, one can prove in ZF that there is a least cardinal strictly
bigger than !1 . It is called !2 , or @2 . In general, we define:
Definition 5.5. Given an ordinal ↵, @↵ , also denoted !↵ , is the ↵–th
infinite cardinal.
Definition 5.6. Given a cardinal , + , the successor of , is the least
cardinal strictly bigger than .
Hence, (@0 )+ = @1 , (@1 )+ = @2 , and in general, (@↵ )+ = @↵+1 .
Proposition 5.7. (ZF) For every infinite cardinal , + exists.
Proof. Similar to the proof that @1 exists: Say that  ✓  ⇥  is a
well–order if (, ) is a well–order on .
Every initial segment of a well–order on  of order type at least  is
order–isomorphic to a well–order on  ([Exercise]). Hence,
=  [ {↵ : ↵ = ot((, )) for some well–order  on }
is transitive and is a set since it is [range(F ), where F : P(⇥) !
Ord is the function sending X to ot((, X)) if X is a well-order on ,
and to 0 otherwise. Hence is an ordinal.
is not bijectible with : If f :
!  were a bijection,
{(↵, ↵0 ) 2  ⇥  : f
1
would be a well–order of . But then
ordinals.
It is easy to see that the ordinal
that is,
(↵) 2 f
1
(↵0 )}
2 , which is impossible for
⇤
in the above proof is in fact + ;
+ =  [ {↵ : ↵ = ot((, )) for some well–order  on }
[Exercise]
Theorem 5.8. (Cantor–Bernstein–Schröder Theorem) (ZF) For all
sets X and Y , the following are equivalent.
28
D. ASPERÓ
(1) |X|  |Y | and |Y |  |X|.
(2) |X| = |Y |
Proof. The implication from (2) to (1) is of course trivial, so we only
need to prove that (1) implies (2). For this, let f : X ! Y and
g : Y ! X be injective functions. By replacing if necessary X and
Y by, for example, X ⇥ {0} and Y ⇥ {1}, respectively, we may assume
that X and Y are disjoint in the first place. Given c 2 X [ Y , let
c be the ✓–maximal sequence with domain included in Z such that
c (0) = c, c (z + 1) = f ( c (z)) or c (z + 1) = g( c (z)) depending on
whether c (z) 2 X or c (z) 2 Y , such that c (z 1) = c̄ if c̄ 2 X is
such that f (c̄) = c (z) (if c (z) 2 Y and if there is such a c̄), and such
that c (z 1) = c̄ if c̄ 2 Y is such that g(c̄) = c (z) (if c (z) 2 X
and if there is such a c̄). We will call c the orbit of c. We say that
c starts in X if there is some z 2 Z in the domain of c such that
c (z) 2 X and such that there is no c̄ 2 Y with g(c̄) = c (z) (so c (z)
is the first member of c ). Similarly we define ‘ c starts in Y ’. And
we say that c does not start in the remaining case (i.e., if and only if
dom( c ) = Z). We also say that a set is an orbit if is (the range
of) the orbit of some c 2 X [ Y in the above sense.
The first observation is that every two distinct orbits are disjoint and
that the orbits partition X [ Y . The second observation is that if is
an orbit, then
• f
is a bijection between \ X and \ Y if starts in X,
• g
is a bijection between \ Y and \ X if starts in Y ,
and
• f
is a bijection between \ X and \ Y and g
is a
bijection between \ Y and \ X if does not start.
Using these two observations we can now define a bijection h : X !
Y by ‘gluing together’ suitable restrictions of f and/or of the inverse
of g: Given a 2 X, if the unique orbit to which a belongs starts in X
or does not start, then h(a) = f (a). And if this orbit starts in Y , let
h(a) be the unique b 2 Y such that g(b) = a.
⇤
As we have seen in this proof, if f : X ! Y and g : Y ! X
are injective functions, then there is a bijection h : X ! Y that
can be e↵ectively constructed from f and g. For example, let f be
the identity on {2n : n 2 !} and let g : ! ! {2n : n 2 !}
be given by g(n) = 4n. These are injective non–surjective functions
between ! and {2n : n 2 !}, and the above proof produces a bijection
MAGIC Set Theory lecture notes (Spring 2014)
29
h : ! ! {2n : n 2 !} that can be e↵ectively constructed from f and
g.
Let us consider the following statement:
Dual C–B–S : For all sets X, Y , the following are equivalent:
(1) |X| = |Y |
(2) There is a surjection f : X ! Y and there is a surjection
g : Y ! X.
Proposition 5.9. (ZFC) Dual C–B–S is true.
Proof. Suppose (2) holds. Using AC we find functions f¯ : Y ! X
and ḡ : X ! Y as follows:
• For every b 2 Y , f¯(b) is some a 2 X such that f (a) = b.
• For every a 2 X, ḡ(a) is some b 2 Y such that g(b) = a.
Then f¯ and ḡ are injective functions, so by C–B–S, |X| = |Y |.
⇤
The following question is apparently open.
Question 5.10. It is not known whether or not, modulo ZF, Dual
C–B–S is equivalent to the Axiom of Choice.
Definition 5.11. A set X is countable if and only if |X| = @0 . A set
is uncountable if it is not finite or countable.
Let us see some examples of countable sets.
Proposition 5.12. (ZF) The following sets are countable.
• ! ⇥ !; in general, n ! := {s : s : n ! !} for any n 2 !,
n
n 1. (the
denoted ! n .)
S setn ! is often<!
<!
• ! := n2! !. (the set ! is also denoted ! <! .)
• [!]n := {X ✓ ! : |X| = n} for any n 2 !, n 1.
• [!]<! = {X ✓ ! : X finite}
• Z
• Q
• The set of algebraic numbers (x 2 C is algebraic if and only if
it is a root of a polynomial with rational coefficients).
Proof. We can produce the corresponding bijections h : X ! ! (or
h : ! ! X) by showing that there are one–to–one functions f : X !
! and g : ! ! X and then appealing to C–B–S. In many cases the
existence of at least one of these one–to–one functions is immediate.
An injective f : ! ⇥ ! ! ! is given for example by f (n, m) =
2n+1 3m+1 (we used this coding in the proof of Proposition 5.4). The
existence of a bijection between n ! and ! can be proved by induction on
n since |n+1 !| = |(n !) ⇥ !|. This given a definable sequence (fn )1n<!
30
D. ASPERÓ
where fn : ! ! n ! is a bijection for all n. We can then find a
bijection f : ! ⇥ ! ! <! ! by, for example, sending (0, 0) to ; and
sending (n, m) 6= (0, 0) to fn+1 (m). Since there is a bijection g : ! !
! ⇥ !, the composition f g : ! ! <! ! is a bijection. To see that
there is a bijection hn : [!]n ! ! for every n
1, send x 2 [!]n
to fn 1 ((x0 , . . . , xn 1 )), where (x0 , . . . , xn 1 ) is the strictly increasing
enumeration of x. We can also code all (the inverses of) these bijections
together into a bijection h : ! ! [!]<! exactly as in the proof of
|<! !| = |!|.
Using the above bijections and any of the usual representations of Z
and Q (as, say, pairs of natural numbers and pairs of integers, respectively), we can easily build bijections between ! and Z and between
! and Q. Using also the above bijections, we can well–order all polynomials with coefficients in Q in length !. Once this is done, we can
easily find a bijection between a subset of ! ⇥! and the set of algebraic
numbers (which gives what we want by C–B–S): Given (n, k), if p(x) is
the n–th polynomial with rational coefficients and p(x) has at least k
distinct roots, then we send (n, k) to the k–th root of p(x) in (say) the
linear order <lex of C given by a0 + ib0 <lex a1 + ib1 i↵ either a0 < a1
or else a0 = b1 and b0 < b1 (where < refers to the usual order on the
real line).
⇤
Proposition 5.13. (ZFC) The union of every countable collection of
countable sets
S is countable: If (Xn )n2! is such that each Xn is countable, then n2! Xn is countable.
S
Proof. @0 S
 | n2! X0 | is clear: There is a bijection f : ! ! X0 , and
f :!
S ! n Xn is an injection.
| n Xn |  @0 : For every n < ! pick, using the Axiom of Choice, a
bijection fn : Xn ! ! (i.e., let X = {Fn : n 2 !} where for each n,
Fn is the set of all pairs (n, f ), where f : Xn ! ! is a bijection, let
G be a choice function
for X, and let fn = f if G(Fn ) = (n, f )).
S
Now let F : n2! Xn ! ! ⇥ ! be the function sending x to
(n, fn (x)) if n is first k < ! such that x 2 Xk . F is an injection,
and if g : ! ⇥
S! ! ! is a bijection (which exists since |! ⇥ !| = @0 ),
then g FS: n Xn ! ! is an injection.
S
S
Since | n Xn |  @0 and @0  | n Xn |, by C–B–S we get | n Xn | =
@0 .
⇤
Some form of Choice is necessary in the above proposition. In fact,
this proposition is not necessarily true without the Axiom of Choice: If
ZF is consistent, then there are models of ZF in which !1 is a countable
union of countable sets (!)
MAGIC Set Theory lecture notes (Spring 2014)
31
Let us see some examples of uncountable sets now:
• !1 , and in fact all ordinals ↵ !1 .
• P(!) (by Cantor’s Theorem 1.2).
Proposition 5.14. |R| = |P(!)|. In particular, R is uncountable.
Proof. Let I be the closed–open interval [0, 1) ✓ R. Since of course
|[0, 1)|  |R|, by C–B–S it suffices to show |P(!)|  |[0, 1)| and |R| 
|P(!)|.
✏n
Let f : P(!) ! [0, 1) send X ✓ ! to ⌃n2! 2n+1
, where ✏n = 0 if
n2
/ X and ✏n = 1 if n 2 X (i.e., (✏n )n2! is the characteristic function
of X).
Let h : Q ! ! be a bijection and let g : R ! P(!) send x 2 R to
{h(q) : q < x} (this < is of course the natural order on R).
f and g are injective functions, so by C–B–S, |[0, 1)| = |R| = |P(!)|.
⇤
Remark: Even if |Q| = @0 < |P(!)| = |R|, the rationals are dense in
the reals, i.e., between every two reals there is some (in fact, infinitely
many) rationals (!)
[Exercise: |C| = |P(!)|.] Hence, since the set of algebraic numbers is countable, most complex numbers are transcendental (i.e., non–
algebraic). In fact
|{x 2 C : x transcendental}| = |C| = |R| = |P(!)|
Definition 5.15. (ZFC) If  is a cardinal, |P()| is denoted by 2 .
In particular, |R| = 2@0 . We have seen that @0 < 2@0 (Cantor’s
Theorem), and therefore @1  2@0 by definition of @1 as the least
uncountable cardinal (we need the Axiom of Choice to conclude that
there is an injection from !1 into R; without AC this is not true in
general!). The following is therefore a very natural question.
Question 5.16. Is @1 = 2@0 ? In other words, if X ✓ R is uncountable,
does it follow that |X| = |R|?
This is perhaps the most famous question in set theory. Georg Cantor was certainly obsessed with it, and it is the first question on the
famous list of problems that David Hilbert presented at his address at
the International Congress of Mathematics in 1900 in Paris. We will
see a “solution” later on.
Definition 5.17. Cantor’s Continuum Hypothesis (CH): 2@0 = @1 .
Theorem 5.18. (ZF) The following are equivalent.
32
D. ASPERÓ
(1) AC
(2) The Well–ordering Principle: Every set can be well–ordered.
Proof. Suppose AC holds. Let X be a set. Let f be a choice function
for P(X) \ {;}. We define enumerations (x↵ : ↵ < ) of subsets of X
by recursion on the ordinals in such a way that (x↵ : ↵ < ) = (x↵ :
↵ < 0)
for all < 0 , as follows: Let be an ordinal and suppose
(x↵ : ↵ < ) has been defined. If {x↵ : ↵ < } = X, then we are
done. Otherwise X \ {x↵ : ↵ < } =
6 ;. Set
x = f (X \ {x↵ : ↵ < })
This defines (x↵ : ↵ <
+ 1). If is a limit ordinal we let
[
(x↵ : ↵ < ) = {(x↵ : ↵ < 0 ) : 0 < }
This gives a class–function F from P(X) to the ordinals, sending Y ✓
X to if Y = X \ {x↵ : ↵ < }. Since P(X) is a set, by Replacement
range(F ) is a set of ordinals so it cannot be all of Ord. Hence this
construction has to stop at some point (there must be
such that
X \ {x↵ : ↵ < } = ;). But then {(x↵ , x↵0 ) : ↵ 2 ↵0 2 } is a
well–order of X.
Now assume the Well–ordering Principle. Let
S X be a set consisting
of nonempty sets and let  be a well–order of X. Now, given a 2 X
let f (a) be the –minimal element of a. Then f is a choice function
for X.
⇤
We have used recursion on the ordinals to define (x↵ : ↵ < ) in
the first part. Later we will see that this can be done. Read again this
proof then.
Corollary 5.19. (ZF) The following are equivalent:
(1) AC
(2) Every set is bijectable with a unique ordinal.
Corollary 5.20. (ZF) The following are equivalent:
(1) AC
(2) (Trichotomy for sets) Given any two sets X, Y , exactly one of
the following holds.
• |X| = |Y |
• |X| < |Y |
• |Y | < |X|
Thus, the Axiom of Choice has the following counterintuitive consequence: R, and even C, can be well–ordered (of course in length 2@0 ).
This well–order has to be highly non–constructive. In fact there are
MAGIC Set Theory lecture notes (Spring 2014)
33
models of ZF in which such a well–order does not exist. The fact that
R can be well–ordered enables one to construct rather pathological
objects (Banach–Tarski decompositions, etc.).
Example: A non–Lebesgue measurable set: Let ⌘ be the equivalence
relation on (0, 1) ✓ R given by x ⌘ y if and only if x y 2 Q. Let f
be a choice function for the quotient set R/ ⌘ (i.e., f picks an element
out of each equivalence class of ⌘). Then the range(f ) is not Lebesgue
measurable. It’s called a Vitali set.
There are extensions of ZF incompatible with AC and which rule out
such pathological consequences of AC as the Banach–Tarski “paradox”,
non–Lebesgue measurable sets, etc. These extensions of ZF say that all
sets of reals have nice regularity properties and therefore seem to reflect
better our intuitions about such sets than ZFC. One such extension is
ZF + “The Axiom of Determinacy”. Sufficiently strong large cardinal
axioms (these are natural axioms extending ZFC, and in fact the Axiom
of Infinity can be regarded as one such axiom) actually imply
L(R) |= ZF + “The Axiom of Determinacy”,
where L(R) is the minimal inner model of ZF containing all the ordinals
and all the reals.
We have seen that |!| < |P(!)|. This implies in particular that a
collection of pairwise disjoint subsets of ! has to be finite or countable.
Definition 5.21. Two sets X, Y are almost disjoint if X \ Y is finite.
Theorem 5.22. (ZF) There is a collection A ✓ P(!) of size 2@0 consisting of pairwise almost disjoint sets.
Proof. Let <! 2 be the complete binary tree of height !, that is, the
tree of n–sequences of 0’s and 1’s, for n < !. By what have seen, <! 2
is countable. In fact, there is a simple enumeration of the nodes of T ,
where the first member is ;, the next two members are h0i and h1i, the
next fours members are h0, 0i, h0, 1i, h1, 0i and h1, 1i, the next eight
members are the sequences with exact three members, and so on. Let
f : ! ! <! 2 be such a bijection.
Now consider any two distinct infinite branches b, b0 through <! 2
and note that b \ b0 is finite; in fact, if n is the first position where they
disagree, then they have all nodes b k, for k  n, in common, but have
no other nodes in common. Also, there are as many infinite branches
through <! 2 as there are subsets of !. In fact, there is obviously a
bijection g from P(!) into the set of such branches sending X ✓ !
to the characteristic function X (i.e., the function sending n to 1 if
n 2 X and to 0 if n 2
/ X).
34
D. ASPERÓ
It follows that {g 1 [b] : b an infinite branch through <! 2} is a subset of P(!) consisting of 2@0 –many pairwise almost disjoint sets.
⇤
6. Foundation, recursion and induction. The cumulative
hierarchy
We have seen recursive definitions, for example when we proved
|n !| = |!| for all n 2 !, n
1 (this was a recursion on !), or when
we proved The Well–ordering Principle from the Axiom of Choice (this
was a recursion on the ordinals).
Also, many familiar definitions are by recursion: For example n! is
defined by
• 0! = 1
• (n + 1)! = n!(n + 1)
Another example: For a given n 2 !, we can define the function
f : ! ! ! given by f (x) = n + x (in other words, we can define
n + m) as follows:
• n+0=n
• n + (m + 1) = (n + m) + 1
In fact, for a given ordinal ↵, we define ↵ + by recursion on the
ordinals by:
• ↵+0=↵
• ↵+S( ) = S(↵+ ) = (↵+ )+1 (recall: +1 = S( ) = [{ }
by definition).
S
• ↵ + = {↵ + : < } if is a nonzero limit ordinal.
Note: + is not commutative: ! + 1 6= ! = 1 + ! !
Two more examples: @↵ is defined, by recursion on the ordinals, by
• @0 = !
+
• @S(↵) (=
S @↵+1 ) = (@↵ )
• @ = {@ : < } if is a nonzero limit ordinal.
In ZFC, for every cardinal , i↵ () is defined, by recursion on the
ordinals, by
• i0 () = 
i↵ ()
• i↵+1 () =
(= |P(i↵ ())|)
S2
• i () = {i () : < } for every nonzero limit ordinal .
Notation: If  = !, we write i↵ for i↵ ().
One last example:
Definition 6.1. We define (V↵ : ↵ 2 Ord) as follows:
• V0 = ;
• V↵+1 = P(V↵ )
MAGIC Set Theory lecture notes (Spring 2014)
• V =
S
{V :
< } if
35
is a limit ordinal.
(V↵ : ↵ 2 Ord) is called the cumulative hierarchy.
•
•
•
•
•
•
•
•
•
•
•
•
•
V0 = ;
V1 = {;} = 1
V2 = {;, {;}} = 2
V3 = {;, {;}, {{;}}, {{;, {;}}}
|V4 | = 24 = 16
|V5 | = 216 = 65536
|V6 | = 265536 (which, according to Wikipedia, is much bigger
than the number of atoms of the observable universe!)
65536 )
|V7 | = 2(2
...
|V! | = @0
|V!+1 | = 2@0 = i1
|V!+2 | = 2i1 = i2
For every ordinal ↵, |V!+↵ | = i↵ .
Also: We prove things by induction on the ordinals: Let P (x) be a
first-order property. Suppose the following.
(1) P (0) holds.
(2) For every ordinal ↵ > 0, if P ( ) holds for every ordinal
then P (↵) holds.
< ↵,
Then P (↵) holds for every ordinal ↵.
Example: ⌃kn k =
Another example:
n(n+1)
2
for every n < !.
Proposition 6.2. (ZF) For every ordinal ↵, V↵ is transitive.
Proof. V0 = ; is transitive.
Let ↵ > 0 be an ordinal and suppose V is transitive for every < ↵.
Suppose ↵ is a successor ordinal, ↵ = + 1. Then V↵ = V +1 = P(V ).
Let x 2 V↵ and let y 2 x. Then x ✓ V . It follows that y 2 x ✓ V
and therefore y 2 V . Since V is transitive by induction hypothesis,
y ✓ V . But then y 2 P(V ) = V↵ .
S
Finally suppose ↵ > 0 is a limit ordinal. Then V↵ =
<↵ V . Let
¯
y 2 x 2 V↵ . Then there is some < ↵ such that x 2 V ¯. S
Since V ¯ is
transitive by induction hypothesis, y 2 V ¯. But then y 2
<↵ V =
V↵ .
⇤
Why is it ok to make definitions by recursion on the ordinals and to
prove things by induction on the ordinals?
36
D. ASPERÓ
Induction is easy: Note that if X is a nonempty class of ordinals,
then min(X) exists. Now, suppose P (x) is a first–order property such
that for every ordinal, if P ( ) holds for all < ↵, then P (↵). We want
to see that P (↵) for all ordinals ↵. Suppose towards a contradiction
that
X = {↵ 2 Ord : ¬P (↵)} =
6 ;
Let ↵ = min(X). For every ordinal < ↵, 2
/ X by definition of
min(X). But then P ( ). Hence we have that P ( ) for all
2 ↵.
Therefore P (↵) by our assumption. So ↵ 2
/ X. This is a contradiction
since ↵ = min(X) 2 X.
What about definitions by recursion?
Theorem 6.3. (ZF) (Recursion (meta)–theorem)
Let G(x, y) be a class–function. Then there is a unique class function
F defined on Ord such that for every ordinal ↵,
F (↵) = G(↵, F
↵)
Proof. We prove, by induction on the ordinals, that for every ordinal ↵
there is a unique function f with domain ↵ such that for every < ↵,
f ( ) = G( , f
)
We show uniqueness first and then existence.
Uniqueness: Suppose f0 and f1 are distinct functions with dom(f0 ) =
dom(f1 ) = ↵ such that f0 ( ) = G( , f0
) and f1 ( ) = G( , f1
)
for every < ↵. Since f0 6= f1 , let
Then f0
= f1
= min{ 2 ↵ : f0 ( ) 6= f1 ( )}
. But then
f0 ( ) = G( , f0
) = G( , f1
) = f1 ( )
Contradiction.
Existence: Let ↵ be an ordinal. For every < ↵ let f be the unique
function h with domain such that h( ) = G( , h
) for all <
(which exists by induction hypothesis).
One can easily prove by induction on < ↵ that if 0 < , then
0
0
f =f
([Exercise]).
S
If ↵ is a limit ordinal, then f ↵ = {f :
< ↵} is as desired by
the previous line.
If ↵ = ↵
¯ + 1, let
f = f ↵¯ [ {(¯
↵, G(¯
↵, f ↵¯ ))}
MAGIC Set Theory lecture notes (Spring 2014)
Let now
< ↵. If
<↵
¯ , then
f ( ) = f ↵¯ ( ) = G( , f ↵¯
If
37
) = G( , f
)
=↵
¯ , then
↵
¯
since f = f
f (¯
↵) = G(¯
↵, f ↵¯ ) = G(¯
↵, f
↵ by definition of f .
↵
¯)
⇤
Example: The class–function F sending ↵ 2 Ord to V↵ is such that
F (↵) = G(↵, F ↵), where G(x, y) is:
• ; if x = 0 or if x is not an ordinal.
• S
P(y(x̄)) if x is the successor ordinal x̄ + 1.
• range(y) if x is a nonzero limit ordinal.
We have seen, by induction on the ordinals, that V↵ is transitive for
every ordinal ↵.
Proposition 6.4. For all ↵ < , V↵ ✓ V .
Proof. Again by induction on . This is vacuously true for = 0. For
a nonzero limit ordinal, V ◆ V↵ by definition of V . For = ¯ + 1,
V = P(V ¯). If ↵ = ¯, then we are done since every member of V ¯
is a subset of V ¯ (as V ¯ is transitive) and therefore V ¯ ✓ P(V ¯). If
↵ < ¯, then V↵ ✓ V ¯ by induction hypothesis. But V ¯ ✓ P(V ¯) by the
previous case, and hence V↵ ✓ P(V ¯) = V .
⇤
S
Definition 6.5. For every x 2 ↵2Ord V↵ ,
rank(x) = min{↵ 2 Ord : x 2 V↵+1 }
Definition 6.6.
S For every set x, the transitive closure of x, denoted
by T C(x), is {Xn : n < !} where
• X0 = x S
• Xn+1 = Xn
S
SS
SSS
So T C(x) = x [ x [
x[
x [ ...
[Exercise: T C(x) is theT✓–least transitive set y such that x ✓ y. In
other words, T C(x) = {y : y transitive, x ✓ y}.]
S Let us fix some notation now for the set–theoretic universe and for
↵2Ord V↵ .
Definition 6.7. V denotes the class of all sets; that is, V = {x : x =
x}.
S
Definition 6.8. WF = {V↵ : ↵ 2 Ord}: The class of all x such
that x 2 V↵ for some ordinal ↵.
38
D. ASPERÓ
In the above definition, WF stands for “well–founded”.
Note: WF is a transitive class: y 2 x 2 V↵ implies y 2 V↵ since V↵
is transitive.
Theorem 6.9. (ZF) V = WF
Proof. Suppose, towards a contradiction, that there is some set x such
that x 2
/ WF. Let y = T C(x).
y2
/ WF: Suppose y 2 V↵ . Since x ✓ y ✓ V↵ (where y ✓ V↵ is true
by transitivity of V↵ ), x 2 P(V↵ ) = V↵+1 . Contradiction.
By Foundation we may find a 2 y, a 2-minimal in y, such that
a 2
/ WF. For every z 2 a, it follows that z 2 y (by transitivity
of y) and therefore z 2 V↵ for some ↵ 2 Ord. Hence, the function
rank a sending z 2 a to rank(z) is defined for all z 2 a. But then
range(rank a) has to be a set by Replacement and therefore there is
some ordinal ↵
¯ such that ↵
¯ > rank(z) for every z 2 a. [No set X of
ordinals can be cofinal in Ord (i.e., such that forSevery ↵ 2 Ord there
is some 2 X with ↵ < ).SWhy? Otherwise X = Ord, which is
not a set (Burali–Forti), but X is a set if X is a set by Union Axiom.
Contradiction.] It follows that for every z 2 a there is some < ↵
¯
such that z 2 V ✓ V↵¯ . Hence a ✓ V↵¯ and therefore a 2 P(V↵¯ ) = V↵+1
¯ .
Contradiction with a 2
/ WF.
⇤
The fact that V = WF realises the idea that a set is any collection
built out of sets already built. This is known as the iterative conception
of sets. Note that this conception of sets rules out such “sets” as V
or the Russell class {x : x 2
/ x}: They couldn’t possibly be sets since
one needs to refer to the totality of sets for their definition, a totality
to which they would belong if they were sets. Take for example, V.
Certainly, if V is a set, then V 2 V. But this goes against the iterative
conception of set, whereby a set is built up out of previously built sets.
7. Inner models and relativization
Let (M, 2M ) be a submodel, or inner model, defined by a formula
⇥(x); in other words, M = {a : ⇥(a)} and, for all a, b 2 M , a 2M b
if and only if a 2 b (we usually leave out 2M and write M instead of
(M, 2M )). (Examples: V, WF, L, HOD, ...).
We define the relativization to M of a formula '(~x), to be denoted
M
' (~x), in the following manner.
• (x 2 y)M is x 2 y.
• (x = y)M is x = y.
M
• ('0 _ '1 )M is 'M
0 _ '1 .
M
M
• (¬') is ¬' .
MAGIC Set Theory lecture notes (Spring 2014)
39
• ((8x)('(x̄))M is 8x(⇥(x) ! 'M (x̄)). We may also write something like (8x 2 M )'M (x).
Note: Suppose T is a theory in the language of set theory. Suppose
(N, E) is a structure in the language of set theory, and suppose M is
an inner model in N . Then (N, E) |= M for every 2 T if and only
if (M, E) |= T .
Notation: If (N, E) is a structure in the language of set theory, M
is an inner model defined by a formula ⇥(x) possibly with parameters
(i.e., M = (N, E (M ⇥ M )), where M = {a 2 N : (N, E) |= ⇥(a)}),
and we want / need to emphasise that M is the inner model defined
by ⇥(x) as defined within (N, E), then we often write M N instead of
M.
Example: WFM
Note: For every ordinal ↵, V↵WF = V↵ (here V↵ refers to the set,
definable from the parameter ↵, with the definition that we have seen).
Many facts about the universe V are inherited by reasonable submodels. For example:
Lemma 7.1. Suppose M is a transitive set or a transitive proper class.
Then M |= Axiom of Extensionality.
Proof. Let a, b 2 M and suppose M |= (8x)(x 2 a $ x 2 b) (this of
course is shorthand for
M |= (8x)(x 2 y $ x 2 z)[~a]
where ~a is any assignment sending the variable y to a and the variable
z to b).
This means that a \ M = b \ M . Since M is transitive (in V), every
member of a or of b is a member of M . It follows that a \ M = a and
b \ M = b and therefore a = b. Hence M |= a = b. In sum, M thinks
that for all y, z, if y and z have the same elements, then they are equal.
In other words, M |= Axiom of Extensionality.
⇤
Also:
Lemma 7.2. Suppose M is a transitive set or a transitive proper class
which is closed under unordered pairs (meaning that for all a, b 2 M ,
{a, b} 2 M ). Then M |= Axiom of Unordered pairs.
Proof. Let c = {a, b} 2 M . Check, as in the previous proof, that
M |= (8x)x 2 c $ x = a _ x = b.
⇤
Similarly:
40
D. ASPERÓ
Lemma S
7.3. Suppose M is a transitive set or a transitive proper class.
Suppose a 2 M for every a 2 M . Then M |= Union set Axiom.
Lemma 7.4. Suppose M is a transitive set or a transitive proper class.
Suppose for every a 2 M there is some b 2 M such that b = P(a) \ M .
Then M |= Power set axiom.
[Proofs: Exercises.]
Note: There are situations in which there are transitive models M
of fragments of ZFC, or even of all of ZFC, and some a 2 M such that
P(a)M is strictly included in P(a) (i.e., there are subsets b of a such
that b 2
/ M ).
Lemma 7.5. Suppose M is a transitive set or a transitive proper class.
If ! 2 M , then M |= Infinity.
Proof idea: As in the previous proofs. The point is that M recognises ; correctly, recognises correctly that something is an ordinal, and
recognises correctly that something is the successor of an ordinal.
We say that the notion of ordinal is absolute with respect to transitive models. It is possible to identify large families of properties that
are absolute with respect to transitive models by virtue of their being
definable by syntactically ‘simple’ formulas (from the point of view of
their quantifiers). We don’t need this kind of general analysis at the
moment so we won’t go into that now.
Note: The notion of finiteness is also absolute with respect to transitive models but, on the other hand, the notion of countability is highly
non–absolute with respect to transitive models: There are transitive
models M and a 2 M such that
M |= a is uncountable
but there is a bijection f : ! ! a, so a is countable in V. The
problem of course is that f is not in M . We will soon see that there
are transitive models of (fragments of) ZFC such that all their sets are
countable in V. And even the whole model can be countable in V.
The notion of choice function is also absolute with respect to transitive models: If M is transitive, a 2 M consists of nonempty sets,
f 2 M , and f is a choice function for M , then we have that M |=
“f is a choice function for a”. Hence:
Lemma 7.6. Let M be a transitive set or a transitive proper class.
Suppose for every a 2 M consisting of nonempty sets there is a choice
function f for a, f 2 M . Then M |= AC.
MAGIC Set Theory lecture notes (Spring 2014)
41
Lemma 7.7. Let M be a transitive set or a transitive proper class.
Suppose b 2 M whenever a 2 M and b ✓ a is definable over M ,
possibly from parameters (in other words, b = {c : c 2 a, M |= '(c, p~)}
for some parameters p~ 2 M ). Then M |= Separation.
Lemma 7.8. Let M be a transitive set or a transitive proper class.
Suppose F [a] 2 M whenever a 2 M , and F is a class–function over M
(in other words, if F is definable by a formula '(x, y, ~z ) which, over M
is functional, p~ 2 M , and a 2 M , then {c : (9b 2 a)M |= '(b, c, p~)} 2
M ). Then M |= Replacement.
7.1. Our first relative consistency proof: Con(ZF \{Foundation})
implies Con(ZF).
Theorem 7.9. Let M |= ZF \{F oundation}. Then M |=
every 2 ZF. Hence, WFM |= ZF.
WFM
for
Proof. By the previous lemmas and the construction of (V↵ : ↵ 2 Ord),
WFM |= for every axiom of ZF \{Foundation} [go through these
axioms one by one them and check that WF is closed under the relevant
operation, then apply the relevant lemma].
To see that M |= FoundationWF holds, let us work in M : Let a 2 V↵ ,
let Z ✓ a, Z 2 WF, and let b 2 Z such that rank(b) = min{rank(z) :
z 2 Z}. Then
WF |= rank(b) = min{rank(z) : z 2 Z}
by absoluteness of the relevant notions. Hence, WF thinks that the
restriction of 2 to a is well–founded. Since this is true for all a 2 WF,
WF |= Foundation
⇤
Corollary 7.10. If ZF \{Foundation} is consistent, then ZF is consistent.
Proof. Suppose ZF \{Foundation} is consistent. By the completeness
theorem we may find a model M |= ZF \{Foundation}. Let M 0 =
WFM . By the theorem M 0 |= ZF. Hence, ZF has a model and therefore
it is consistent.
⇤
Remark 7.11. By exactly the same argument, if M is a model of
M
ZFC \{F oundation}, then M |= WF for every 2 ZFC. Hence,
Con(ZFC \{Foundation}) implies Con(ZFC).
Similar relative consistency results: One can define “the constructible
universe” L:
42
D. ASPERÓ
• L0 = ;
• L↵+1 = Def(L↵ ), where Def(L↵ ) is the set of all subsets of L↵
definable over L↵ possibly with parameters, i.e., the collection
of all sets of the form
{b 2 L↵ : L↵ |= '(b, a0 , . . . , an 1 )}
for some
S formula '(x, ~x) and a0 , . . . , an 1 2 L↵ .
• L = ↵< L↵ if > 0 is a limit ordinal.
S
L = ↵2Ord L↵ .
This construction is due to Gödel. He proved that if we do this construction in ZF, then L |= ZF but also L |= AC and L |= CH.
The above results imply that if ZF is consistent, then ZFC is also
consistent, and in fact also ZFC+CH. Linking this to the implication
we have seen we thus have that if ZF \{Foundation} is consistent, then
so is ZFC+CH.
These relative consistency proofs proceed by building suitable inner
models.6 From now on we will mostly aim at “extending” models of
given models of (fragments) of ZFC (I believe). But first we will need
a string of preliminaries. These preliminaries are in fact very central
notions in set theory and other areas of mathematical logic.
8. Elementary substructures and Skolem closures
Let M and N be two structures (in a first order language L) and let
j:M !N
We say that j is an elementary embedding if and only if for every
L–formula '(x0 , . . . , xn 1 ) and all a0 , . . . , an 1 2 M ,
if and only if
M |= '(a0 , . . . , an 1 )
N |= '(j(a0 ), . . . , j(an 1 ))
Note: If f : M ! N is an isomorphism, then f is an elementary
embedding between M and N . 7 The converse is not true: There are
elementary embeddings j : M ! N such that j is not an isomorphism.
6What
set–theorists understand by “inner models” are usually much more complicated than WF or L. However, the construction of L is in fact the paradigm for
most of these more complicated constructions.
7This is of course the reason why people are interested in structures modulo their
isomorphism types (i.e., two isomorphic structures are regarded as being essentially
the same structure).
MAGIC Set Theory lecture notes (Spring 2014)
43
If M ✓ N and the identity on M is an elementary embedding from
M to N , we say that M is an elementary substructure of N and
write
M 4N
For example: If M ✓ N , a 2 M ,
but
M |= a is uncountable,
N |= a is countable,
then M is certainly a substructure of N but it is not an elementary
substructure of N .
Given a first order language L, we assign a rank to L–formulas in
the following way: We say that
• an atomic formula has rank 0,
• if ' has rank n, then ¬' has rank n + 1,
• if '0 has rank n0 and '1 has rank n1 , then '0 _ '1 has rank
max{n0 , n1 } + 1, and
• if '(x, ~x) has rank n, then (9x)('(x, ~x)) has rank n + 1.
Thus, the rank of a formula measures its complexity.
Lemma 8.1. (Tarski–Vaught Lemma) Let L be a first order language,
let M ✓ N be L–structures, and suppose for all a0 , . . . , an 1 2 M and
every L–formula '(x, x0 , . . . , xn 1 ), if
N |= (9x)'(x, a0 , . . . , an 1 ),
then there is some a 2 M such that
Then
N |= '(a, a0 , . . . , an 1 )
M 4N
Proof. We prove, by induction on the complexity of L–formulas, that
the conclusion of the lemma holds for every L–formula. For atomic
formulas there is nothing to prove.
Suppose ' = ¬'0 . Given a0 , . . . , an 1 2 M , we want to see that M |=
¬'0 (a0 , . . . , an 1 ) i↵ N |= ¬'0 (a0 , . . . , an 1 ). But we have that M |=
¬'0 (a0 , . . . , an 1 ) i↵ M 6|= '0 (a0 , . . . , an 1 ) i↵ N 6|= '0 (a0 , . . . , an 1 )
(by induction hypothesis) i↵ N |= ¬'0 (a0 , . . . , an 1 ).
Now suppose ' = '0 _ '1 . Given a0 , . . . , an 1 2 M , we want to
see that “M |= '0 (a0 , . . . , an 1 ) or M |= '1 (a0 , . . . , an 1 )” holds i↵
“N |= '0 (a0 , . . . , an 1 ) or N |= '1 (a0 , . . . , an 1 )” holds. But “M |=
'0 (a0 , . . . , an 1 ) or M |= '1 (a0 , . . . , an 1 )” holds if and only if “N |=
44
D. ASPERÓ
'0 (a0 , . . . , an 1 ) or N |= '1 (a0 , . . . , an 1 )” holds (by induction hypothesis with '0 and '1 , resp.), so we are done.
Finally suppose ' = (9x)('0 (x, ~x)). Given a0 , . . . , an 1 2 M , we
want to see that there is some a 2 M such that M |= '0 (a, a0 , . . . an 1 )
if and only if there is some a 2 N such that N |= '0 (a, a0 , . . . , an 1 ).
If there is some a 2 M such that M |= '0 (a, a0 , . . . , an 1 ), then N |=
'0 (a, a0 , . . . , an 1 ) by induction hypothesis with '0 . Finally, if there
is some a 2 N such that N |= '0 (a, a0 , . . . , an 1 ), then there is some
a 2 M such that N |= '0 (a, a0 , . . . , an 1 ) by our assumption. But then
M |= '0 (a, a0 , . . . an 1 ) again by induction hypothesis with '0 .
⇤
Definition 8.2. Suppose '(x, x0 , . . . , xn 1 ) is a formula, M is a structure, and F : n M ! M . F is a Skolem function for ' if for all
a0 , . . . , an 1 2 M , if there is some a 2 M such that
M |= '(a, a0 , . . . , an 1 ),
then
M |= '(F (a0 , . . . , an 1 ), a0 , . . . , an 1 )
That is, F chooses a witness to the existential statement
(9x)'(x, a0 , . . . , an 1 )
whenever this is possible.
Sometimes we denote a function as in the above definition by F'M .
Working in ZFC (so, with Choice):
Let L be a first order language and let FmlL denote the set of L–formulas. Suppose N is an
L–structure and X ✓ N . Our next goal will be to build a “small” M
such that
• X ✓ M ✓ N and
• M 4N
Small will mean |M | = |X [ FmlL |.
The construction: Let {F'N : '(x, x0 , . . . xn 1 ) 2 FmlL } be a
collection of Skolem functions for N ; in other words, for every formula '(x, x0 , . . . xn 1 ) and for all a0 , . . . , an 1 2 N , if we have N |=
(9x)'(x, a0 , . . . , an 1 ), then N |= '(F'N (a0 , . . . , an 1 ), a0 , . . . , an 1 ).
For every formula '(x, x0 , . . . xn 1 ) there is some Skolem function as
above by the Axiom of Choice: Simply pick a choice function of F ' ,
where
F ' = {Fa'0 ,...,an
and where
Fa'0 ,...,an
1
1
: a0 , . . . , a n
1
2 N, Fa'0 ,...,an
1
6= ;},
= {b 2 N : N |= '(b, a0 , . . . , an 1 )}
MAGIC Set Theory lecture notes (Spring 2014)
45
for all a0 , . . . , an 1 2 N .
We can then of course pick one Skolem function for each ' by using
again AC and that gives the existence of
{F'N : '(x, x0 , . . . xn 1 ) 2 FmlL }
The idea is to take M to be the “closure of X under Skolem functions.” This closure will be of size |X [FmlL | and will be an elementary
substructureSof N by Tarski–Vaught:
Let M = i2! Xi , where
• X0 = X,
• for all i < !,
Xi+1 = Xi [ {F'N (~a) : ~a 2
<!
Xi , '(x, ~x) 2 FmlL }
S
Clearly Xj ✓ Xj 0 ✓ M for all j < j 0 < ! and X ✓S i<! Xi = M .
Also, for every '(x, ~x) 2 FmlL and all a0 , . . . , an 1 2 i<! Xi , if N |=
(9x)'(x, a0 , . . . , an 1 ), then N |= '(F'N (a0 , . . . , an 1 ), a0 , . . . , an 1 ).
Let i⇤ < ! be such that a0 , . . . , an 1 2 Xi⇤ . This i⇤ exists since
aj 2 Xij for all j < n and since Xj ✓ Xj 0 for all j < j 0 < !. But then
F'N (a0 , . . . , an 1 ) 2 Xi⇤ +1 ✓ M . Hence, by the Tarski–Vaught Lemma,
M 4 N.
Smallness of M : Let us prove the following special case (which will
be enough for our purposes):
Claim 8.3. Suppose |L|  @0 and |X|  @0 . Then | FmlL | = |M | =
@0 .
Proof. Note: By |L|  @0 we may identify L with a subset of !. Then
we may code L–formulas in some natural way by members of <! !. But
|<! !| = @0 .
We prove by induction on i < ! that |Xi |  @0 : True for i = 0 by
assumption.
Let i < ! and suppose |Xi |  @0 . Then
|Xi+1 |  |<! Xi ⇥ FmlL |  |<! ! ⇥ !|
since | FmlL | = @0 . But |<! ! ⇥ !| = |<! !| = @0 .SLet fi : Xi ! ! be
an injective function for each i < !. Finally, f : i<! Xi ! ! ⇥ ! is
an injective function,
where f (x) = (i⇤ , fi⇤ (x)) if i⇤ = min{i : x 2 Xi },
S
and therefore | i Xi |  @0 since |! ⇥ !| = @0 .
⇤
We have proved:
Theorem 8.4. (ZFC) (Downward Löwenheim–Skolem Theorem, special case) Let L be a language which is either finite or countable. Let
N be an L–structure and let X ✓ N be such that |X|  @0 . Then there
is some M such that
46
D. ASPERÓ
• X✓M ✓N
• M 4N
• |M |  @0 .
8.1. Reflection. Given formulas 0 , 1 , we say that 0 is a subformula
of 1 if and only if
• 0 = 1 , or
• there is a subformula of 1 such that 0 is a subformula of
, or
• either 1 = ¬ 0 or 1 = 0 _ for some or 1 = _ 0 for
some or 1 = (9x)('0 ) for some variable x.
Theorem 8.5. (ZF) (Reflection) Let '(x0 , . . . , xn 1 ) be a formula in
the language of set theory. There is a proper class C' of ordinals such
that:
(1) for all ↵ 2 C' and all a0 , . . . , an 1 2 V↵ ,
V↵ |= '(a0 , . . . , an 1 )
if and only if '(a0 , . . . , an 1 ) is true (we may also write V |=
'(a0 , . . . , an 1 )). We write V↵ 4' V.
(2) For every 2 Ord there is some ↵ 2 C' such that < ↵ (C'
is unbounded).
(3) For every limit ordinal , if sup(C' \ ) = , then 2 C' (C'
is closed).
Proof. Let ('i : i < n) list all subformulas of ' in such a way that if
'i is a subformula of 'i0 , then i < i0 .
We build a ✓–decreasing sequence C'i of closed and unbounded
proper classes of ordinals (for i < n) as follows.
Let i < n and suppose C'i 1 defined if i > 0. If i = 0 (so 'i is an
atomic formula) then C'i = Ord. If i > 0 but 'i 6= (9x)('i0 ) for any
i0 < i and any variable x, then C'i = C'i 1 .
Finally, suppose i > 0 and 'i = (9x)'(x, x0 , . . . , xm 1 ) for some
i0 < i and some variable x. We define C'i = {↵⇠ : ⇠ 2 Ord}, where
(↵⇠ )⇠2Ord is a strictly increasing and continuous – i.e., at limit stages ⇠
we let ↵⇠ = sup⇠0 <⇠ ↵⇠0 – sequence of ordinals defined as follows:
Given an ordinal ⇠, if ↵⇠ has been defined let 0⇠ = ↵⇠ and let 1⇠ be
the least > ↵⇠ , 2 C'i 1 , such that for all a0 , . . . , am 1 2 V↵⇠ , if
there is some b such that '(b, a0 , . . . , am 1 ), then there is some b 2 V
⇠
such that '(b, a0 , . . . , am 1 ). In general, if n⇠ has been defined, let n+1
be the least > n⇠ , 2 C'i 1 , such that for all a0 . . . , am 1 2 V n⇠ , if
there is some b such that '(b, a0 , . . . , am 1 ), then there is some b 2 V
such that '(b, a0 , . . . , am 1 ).
MAGIC Set Theory lecture notes (Spring 2014)
47
Let
= supn<! n⇠ , and note that
2 C'i 1 since C'i 1 is closed.
Let ↵⇠+1 = .
The construction of ↵0 of course is as above, starting from 0 instead
of ↵⇠ .
Since V↵ 4'i0 V for all ↵ 2 C'i 1 , it follows that V↵ 4'i V for all
↵ 2 C 'i :
Suppose ↵ 2 C'i and a0 , . . . , am 1 2 V↵ . If there is some b 2 V↵ such
that V↵ |= '(b, a0 , . . . , am 1 ), then of course V |= 'i0 (b, a0 , . . . , am 1 )
since V↵ 4'i0 V. And if V |= (9x)'i0 (x, a0 , . . . , am 1 ), then by construction of C'i there is some b 2 V↵ such that V |= 'i0 (b, a0 , . . . , am 1 ).
But then V↵ |= 'i0 (b, a0 , . . . , am 1 ) since V↵ 4'i0 V.
⇤
Corollary 8.6. Suppose '0 , . . . , 'n are finitely many formulas in the
language of set theory. Then there is a proper class C of ordinals such
that:
(1) For every ↵ 2 C and every j  n, V↵ 4'j V.
(2) For every 2 Ord there is some ↵ 2 C such that < ↵.
(3) For every limit ordinal , if sup(C \ ) = , then 2 C.
Proof. Simply note that C'0 \ . . . \ C'n is a closed and unbounded
class of ordinals. It is clearly closed. To see that it is unbounded, given
2 Ord let (↵i )i<! be such that ↵0 > and such that for all i < !
and all j  n there is some 2 C'j , ↵i < < ↵i+1 . Then, for every
j  n, supi<! ↵i is a limit of ordinals in C'j and therefore in C'j . ⇤
8.2.
–systems.
Definition 8.7. A set X is a –system if and only if there is a set R
such that for all a, a0 2 X, if a 6= a0 , then a \ a0 = R.
R is called the root of X.
Theorem 8.8. (ZFC) Let X be an uncountable set such that every
a 2 X is finite. Then there is an uncountable set X 0 ✓ X such that X 0
is a –system.
Proof. We may fix a one–to–one enumeration (a⌫ : ⌫ < !1 ) of (an
uncountable subset of) X.
Let F be a certain finite set of formulas. Exactly which formulas are
in F can be decided by looking at the rest of the proof and identifying
exactly which correctness facts we will be using. By Reflection we may
find an ordinal ↵ such that V↵ is correct about ' for every ' 2 F , i.e.,
for all ' 2 F and Z 2 V↵ , V↵ |= '(Z) if and only V |= '(Z).
48
D. ASPERÓ
Let M be an elementary substructure of (V↵ , 2) such that (a⌫ :
⌫ < !1 ) 2 M and |M |  @0 (M exists by the Löwenheim–Skolem
Theorem we have seen). In fact it will typically be the case that actually
|M | = @0 , for example if F contains a sentence saying that ! exists.
We may assume that M is correct about countable sets, and therefore
x ✓ M for every countable x 2 M : For every x 2 M , x is countable
if and only if M |= x is countable (if F contains “x is countable”).
Since we may assume that ! 2 M and ! ✓ M , if x is countable, then
there is, in M , a bijection f : ! ! x. But then x = f [!] ✓ M . In
particular, M := M \ !1 2 !1 : If ↵ 2 and 2 M \ !1 , then ✓ M
since is countable, and therefore ↵ 2 M . Hence \ M is a countable
transitive set of ordinals and therefore it is an ordinal.
Since X is countable, we may take ⌫ ⇤ 2 !1 \ M .
We may also assume that M |= Axiom of unordered pairs. Hence,
R := a⌫ ⇤ \ M 2 M (since a⌫ ⇤ \ M is a finite subset of M ).
Claim 8.9. For every ⌫¯ 2
R ✓ a⌫ 0 and
M
there is some ⌫ 0 2 !1 , ⌫ 0 > ⌫¯, such that
a⌫ 0 \
[
⌫<¯
⌫
a⌫ ✓ R
Proof. Let ⌫¯ 2 M . TheSstatement “there is some ⌫ 0 2 !1 , ⌫ 0 > ⌫¯, such
that R ✓ a⌫ 0 and a⌫ 0 \ ⌫<¯⌫ a⌫ ✓ R” – with ⌫¯, R and (a⌫ : ⌫ < !1 ) as
parameters – is witnessed by a⌫ ⇤ : Certainly R ✓ a⌫ ⇤ . Also, for every
⌫ < ⌫¯, a⌫ ✓ M , and hence a⌫ ⇤ \ a⌫ ✓ a⌫ ⇤ \ M = R.
But since we may assume M to be correct about this statement, it
holds in M , which gives that M thinks
that there is some ⌫ 0 2 !1 ,
S
⌫ 0 > ⌫¯, such that R ✓ a⌫ 0 and a⌫ 0 \ ⌫<¯⌫ a⌫ ✓ R. But again we may
assume that M is correct about this fact, so in V it S
is true that there
0
0
0
0
is some ⌫ 2 !1 , ⌫ > ⌫¯, such that R ✓ a⌫ and a⌫ \ ⌫<¯⌫ a⌫ ✓ R. ⇤
Claim 8.10. For every ⌫¯ < !1 there is some ⌫ 0 < !1 , ⌫ 0 > ⌫¯, such
that R ✓ a⌫ 0 and such that
[
a⌫ 0 \
a⌫ ✓ R
⌫<¯
⌫
Proof. Suppose otherwise. We may assume that M correctly believes
the negation of the claim. But then there is some ⌫¯ 2 M such that
0
0
M thinks
S that there is no ⌫ < !1 , ⌫ > ⌫¯, such that R ✓ a⌫ 0 and
a⌫ 0 \ ⌫<¯⌫ a⌫ ✓ R. Since we may assume as well that M is correct
about this fact about a⌫¯ , this fact is true in V. But this contradicts
Claim 8.9.
⇤
Using Claim 8.10 we can build a
recursion on !1 :
–system (a⌫i )i<!1 with root R by
MAGIC Set Theory lecture notes (Spring 2014)
49
a⌫0 is any ⌫ such that R ✓ a⌫ (which exists by Claim 2). Given
i, assuming a⌫i0 has been defined for all i0 < i, let ⌫i be the least
⌫ > supi0 <i ⌫i0 such that
• R✓S
a⌫ , and
• a⌫ \ i0 <i a⌫i0 ✓ R.
This ⌫ exists by Claim 2. This completes the proof of the theorem. ⇤
Simplification: It would have been enough to take any ordinal ↵
such that (a⌫ : ⌫ < !1 ) 2 V↵ and then take any countable M 4 V↵
such that (a⌫ : ⌫ < !1 ) 2 M . The reason this suffices is that V↵ is
correct about all the relevant facts about (a⌫ : ⌫ < !1 ).
9. Forcing
Forcing is a method, devised by Paul Cohen (1934–2007) in 1963 in
order to prove relative consistency results in the context of ZF and ZFC.
In other words: For some statement in the language of set theory,
prove Con(ZFC) ! Con(ZFC + ) (and similarly with ZF instead of
ZFC, or with certain extensions of ZFC, etc.). In particular, Cohen
proved
• Con(ZF) ! Con(ZF +¬AC)
• Con(ZF) ! Con(ZFC +2@0 = @2 ), Con(ZF) ! Con(ZFC +2@0 =
@27 ), Con(ZF) ! Con(ZFC +2@0 = @!2 +3456 ), etc.
These results earned him the Fields medal in 1966. The consistency
of ZFC and of ZFC +2@0 = @1 (in other words, ZFC + CH) relative to
the consistency of ZF had been proved already in the late 1930’s by K.
Gödel; in fact he had proved, in ZF, that L |= ZFC + CH.
Forcing has proved to be an extremely powerful method; a cornucopia
of di↵erent models of set theory have appeared using this method in
these 50 years. In fact, forcing is essentially the only method we have
for producing consistency (with ZF or ZFC or ZFC + large cardinals) of
statements with interesting mathematical content (i.e., not ‘artificially
built’ statements as in the proof of Gödel’s incompleteness theorems).
It might even be that there is an actual theorem behind the above
observation, but this is still one of the main open questions in set
theory (known as the ⌦ Conjecture).
The main informal idea of forcing can be described as the wish to
build an extension V[G] of the universe V, where
• V[G] is the “minimal” extension W of V satisfying ZF and
such that G 2 W, and
50
D. ASPERÓ
• G is a set that has been chosen in such a way that V[G] necessarily satisfies some statement of interest (V[G] is forced to
satisfy ).
Strictly speaking this is of course nonsense: V is the universe, it
contains all sets!
One way to make sense of the above idea (not the only way, but
conceptually the cleanest for us now) is the following: Working in V:
We will start with a countable model N of a sufficiently large finite
fragment F of ZFC. Exactly which theory F is will depend on exactly
which facts in the metatheory the argument at hand is going to use.
Our argument will be finite (all arguments are finite), so it will use
a finite set of axioms. We can find such a model N in V, without
assuming Con(T ) in V, thanks to Reflection + Downward Löwenheim–
Skolem (!).
We will then be able to find a transitive model (M, 2) isomorphic to
(N, 2) (M is the transitive collapse of N ). Why?
Say that a relation R is set–like i↵ for all x, {y : yRx} is a set.
Theorem 9.1. (Mostowski) Suppose (X, R) is a well–founded set–like
relation such that (X, R) |= Axiom of Extensionality. Then there is a
unique transitive class (a set or a proper class) M for which there is
an isomorphism ⇡ : (X, R) ! (M, 2).
[Proof: Exercise and/or easy to find.]
Using the theory of forcing we will be able to find a set G (in V, of
course) such that
• M [G] is the ✓–minimal transitive model M 0 such that M ✓ M 0 ,
G 2 M and such that M 0 |= F .
(when building M [G] we say that we have forced over M ). If we are
skilled enough, we will be able to ensure that M [G] |=
for some
statement of interest.
What is this good for?
Now suppose we want to prove Con(ZFC) ! Con(ZFC + ). Assume
Con(ZFC). Suppose, towards a contradiction that there is a proof of
¬ from ZFC.
Let F0 be the finite set of axioms of ZFC involved in this proof. Let
F ◆ F0 , F ✓ ZFC, be still finite and such that the all relevant points in
the theory of forcing that we are about to see can be developed within
F . Since ZFC is consistent, let (M, E) |= ZFC. Working inside (M, E),
argue as before: Find, in (M, E), a countable transitive model M of F
(F can be coded by a natural number in the meta-theory, but (M, E)
contains (interpretations of) all natural numbers from the meta-theory,
MAGIC Set Theory lecture notes (Spring 2014)
51
so F 2 M and F is, in M, a finite set of ZFC axioms). Note that M
is only transitive from the point of view of (M, E), of course.
Suppose in M we can force over M and build M [G] 2 M such that
M [G] |= F + (this last step will obviously depend of the statement
at hand, if is 0 = 1 this of course cannot be done, if it is 2@0 = @27 ,
then yes).
Now we are done: Since F0 ` ¬ , M [G] |= ¬ . But we have shown
M [G] |= . Contradiction.
Note that the above construction scheme does not work to prove
statements of the form Con(ZF) ! Con(ZF + ) in general (i.e., without
AC): One reason is that inside of (M, E) we might not be able in
general to find a countable transitive models M of F . And even if we
could find it, we might in general not be able to find a suitable generic
filter G over M within (M, E) (see Subsection 9.2). There are, however,
other constructions to make things work also in the ZF–context.
9.1. The method of forcing: Partial orders and genericity (an
example). We take a closer look at the idea behind the construction
of forcing extensions of our countable transitive model M . Let us see
a specific example (and the simplest one):
Adding a Cohen real: For us here, a real will be a function f :
! ! 2 (equivalent, a subset of !). We want to add a new real x to M
and build the ✓–minimal extension M 0 such that M ✓ M 0 and x 2 M .
We want to make sure that x is not in M . In particular, for every
real r 2 M there must be some n 2 ! such that x n 6= r n. The
notion of genericity over <! 2 will guarantee that:
Let us consider the partial order C = (<! 2, ) 2 M , where q  p
if and only if p ✓ q (intuitively, q  p means that q “carries more
information about the generic real” (see below)). We say that q is
stronger than p. We call members of C conditions in C and say that C
is a forcing notion.
So C is the partial order consisting of finite sequences of 0’s and 1’s,
and given two conditions p, q in C, q is stronger than p if and only if
q extends p. Reals correspond to infinite branches through C (growing
downwards, that is).
We say that D ✓ <! 2 is dense in C if and only if for every p 2 <! 2
there is some q 2 D such that q  p. For example, for every r : ! ! 2,
r 2 M,
Dr = {p 2
is s dense set.
<!
2 : p 6= r n for some n < !} 2 M
52
D. ASPERÓ
We say that x : ! ! 2 is C–generic over M if and only if the branch
b = {x n : n < !} is such that for every dense D ✓ <! 2, D 2 M ,
b \ D 6= ;. The idea is that x is generic over M if x is “completely
random” from the point of view of M , in the sense that it avoids every
“regularity pattern” specifiable in M . Here is a specific example: the
set
X = {y 2 ! 2 : y(n) = y(n + 1) for every even n < !}
is such that
D = {p 2
<!
2 : p 2 r for some r 2 ! 2 \ X} 2 M
is dense. Hence, the generic real x will be such that x(n) 6= x(n + 1) for
some even n. And similarly, it will also be such that x(n) = x(n + 1)
for some even n. And so on.
Remarkable fact: If x is C–generic over M , M transitive, and
M |= F for a sufficiently large finite fragment F of ZFC, then there is
M [x] such that M [x] is the ✓–minimal M 0 such that
• M 0 is a transitive model of F ,
• M ✓ M 0 , and
• x 2 M 0.
In other words, M 0 is the ✓–minimal obtained from adding the new
real x to M . We call x a Cohen real over M (because it is generic for
C over M and C is known as Cohen forcing).
9.2. Formal development of forcing. A forcing notion is a partial
order (P, P ) with a P –maximum, which we will sometimes denote
1P . (1P is the trivial condition or weakest condition). Members of P
are called P–conditions.
We say that two P–conditions p0 , p1 are compatible if there is a P–
condition p such that p P p0 and p P p1 . Otherwise they are called
incompatible.
(P, P ) is non-atomic if for every p 2 P there are q0 P p and q1 P p
such that q0 and q1 are incompatible.
D ✓ P is dense if and only if for every p 2 P there is some q 2 D,
q P p.
A nonempty set G ✓ P is a filter if the following holds:
• If q 2 G and p 2 P, q P p, then p 2 G.
• For all p1 2 G and p0 2 G there is some p 2 G such that p P p0
and p P p1 .
MAGIC Set Theory lecture notes (Spring 2014)
53
Let M be a model of (enough of) ZFC. Let (P, ) 2 M be a partial
order in M . Let G ✓ P, G 2 V, be a filter. We say that G is P–generic
over M if G \ D 6= ; for every dense set D ✓ P such that D 2 M .
Note: If P is non-atomic and G is P–generic over M , then G 2
/ M:
D = P \ G is dense ([Exercise]). But now, if G 2 M , then D 2 M , and
so G \ D = ;. Contradiction.
Given a transitive M , a forcing notion P 2 M and a P–generic filter
G over M , we will build the forcing extension (or generic extension)
M [G]. M ✓ M [G] and G 2 M [G], so M 6= M [G] if P is non-atomic.
However, M will have a certain amount of access to M [G] (this will be
the reason why M [G] |= F if M |= F ). In fact, members x of M [G]
will have names ẋ in M . x will be the interpretation of ẋ by G:
Given a partial order P, we define the class of P–names by recursion
on the rank, as follows: A set ẋ is a P–name if and only if it is a set of
ordered pairs (p, ẏ) such that
• p 2 P and
• ẏ is a P–name.
For example, ; is the simplest name. And if p, p0 2 P, then both of
⌧ = {(p, ;)} and = {(p, {(p, ;)}), (p0 , ;)} are P–names.
The following names play a distinguished role: Given a forcing notion
P, we define x̌ (for all x), again by recursion on the rank, as follows:
For every x,
x̌ = {(1P , y̌) : y 2 x}
x̌ is called the canonical P–name for x.
Interpreting names: Given a P–name ẋ and a filter G ✓ P, the
interpretation of ẋ by G, denoted valG (ẋ) or ẋG , is
ẋG = {ẏG : there is some p 2 G such that (p, ẏ) 2 ẋ}
For example, the interpretation of the name ; is of course always ;
itself. Now suppose ⌧ and are the above names and G ✓ P is a filter.
If p 2 G, then ⌧G = {;} = 1, but if p 2
/ G, then ⌧G = ; (so di↵erent
names, like ; and ⌧ , can sometimes have the same interpretation).
Also, if p 2 G, then G = {{;}} = {1} if p0 2
/ G and G = {{;}, ;} = 2
if p0 2 G. Finally, if p 2
/ G, then G = ; if p0 2
/ G, and G = {;} = 1 if
0
p 2 G. We thus see in particular that can be interpreted as one of
{1}, 2, 0 and 1.
The canonical name of any x is always interpreted as x:
Lemma 9.2. For every forcing notion P, every set x, and every filter
G ✓ P, x̌G = x.
Proof. By induction on the rank: Suppose y̌G = y for all y 2 x. Then
x̌G = {⌧G : (p, ⌧ ) 2 x̌ for some p 2 G such that (p, ⌧ ) 2 x̌} = {y̌G :
54
D. ASPERÓ
y 2 x}, where this equality is true since necessarily 1P 2 G (as G 6= ;).
But {y̌G : y 2 x} = {y : y 2 x} by induction hypothesis, and
{y : y 2 x} is of course x.
⇤
Now we can define M [G]: Given a transitive model M , a forcing
notion P 2 M , and a P–generic filter G over M ,
M [G] = {ẋG : ẋ 2 M, ẋ a P–name}
Let
Ġ = {(p, p̌) : p 2 P}
Lemma 9.3. For every transitive M , every forcing notion P 2 M and
every P–generic filter G over M , if Ġ is the above name, then ĠG = G.
Proof. ĠG = {p̌G : (p, p̌) 2 Ġ for some p 2 G} = {p : p 2 G} =
G.
⇤
Corollary 9.4. Let M be a transitive model of F , let P 2 M be a
forcing notion, and let G ✓ P be a filter. Then the following hold.
(1) M ✓ M [G] and G 2 M .
(2) M [G] is transitive and Ord \M = M [G] \ Ord.
(3) If N is a transitive model of F and M [{G} ✓ N , then M [G] ✓
N.
Proof. (1): By the first of the above two lemma, for every x 2 M ,
x̌G = x 2 M [G]. By the second one, since Ġ 2 M , ĠG = G 2 M [G].
(2): It ẋ 2 M and y 2 ẋG , then in particular y = ẏG for some P–
name ẏ such that (p, ẏ) 2 ẋ for some p 2 G. But ẏ 2 M since M is
transitive ({{p}, {p, ẏ}} 2 M , which implies {p, ẏ} 2 M , and therefore
ẏ 2 M ). Hence M [G] is transitive.
Ord \M ✓ Ord \M [G] since ǎ 2 M for every ordinal ↵ 2 M and
↵
ˇ G = ↵.
Also: rank(ẋG )  rank(ẋ) for every P–name ẋ [Exercise.] But then, if
↵ = ẋG 2 OrdM [G] for some P–name ẋ 2 M , then ↵  rank(ẋ) 2 OrdM
(because every transitive model of ZF is correct about rank).
(3): Since G 2 N , N can correctly build ẋG from ẋ and G by recursion. Hence ẋG 2 N for every P–name ẋ 2 M .
⇤
We will see that if P 2 M is a forcing notion in M and G is P–generic
over M , then M |= F (i.e., if M satisfies enough of ZFC, then M [G]
satisfies enough of ZFC). But why can we find G?
Since M is countable, we can fix an enumeration (Dn : n < !) of all
dense subsets D of P such that D 2 M (this enumeration is obviously
MAGIC Set Theory lecture notes (Spring 2014)
55
not in M ). Now we can build G recursively, using AC, as follows. Let
(qn )n<! be a P –decreasing sequence such that
• q0 2 D0 , and
• for all n, qn+1 2 Dn+1 , qn+1 P qn .
Given qn , qn+1 may be found since M knows that Dn+1 is dense. The
sequence (qn )n<! is not in M , but each qn is. Finally, let G = {p 2 P :
qn P p for some n}.
9.3. The forcing relation
P.
Definition 9.5. Given a transitive model M of (enough of) ZF, a
forcing notion P 2 M , a formula '(x0 . . . , xn 1 ) in the language of set
theory, P–names ⌧ 0 , . . . , ⌧ n 1 in M , and p 2 P,
p
P
'(⌧ 0 , . . . , ⌧ n 1 )
if and only if for every G, if G is a P–generic filter over M and p 2 G,
then M [G] |= '(⌧G0 , . . . , ⌧Gn 1 ).
We will say that p forces '(⌧ 0 , . . . , ⌧ n 1 ) and will call P the forcing
relation for P.
Strictly speaking we should have included M in the definition (M
will be understood or irrelevant).
We will see that
a certain relation
and will prove
P
⇤
P
is actually definable over M . In fact we will define
over M by recursion on complexity of formulas,
= ⇤P
Given D ✓ P and p 2 P, D is dense below p if and only if for every
q 2 P, if q P p, then there is some r 2 D, r P q.
P
Lemma 9.6. If G ✓ P is P–generic over M , p 2 P, D ✓ P is dense
below p, and D 2 M , then G \ D 6= ;.
[Proof: Exercise.]
Definition 9.7. Let p 2 P.
• Given P–names and ⌧ , p
{r 2 P : r
⇤
P
⇤
P
2 ⌧ i↵
= ⇡ for some (q, ⇡) 2 ⌧ such that r P q}
is dense below p.
• p ⇤P = ⌧ if and only if for all (q, ⇡) 2 [ ⌧ and all r such
that r P p and r P q, r ⇤P ⇡ 2 ⌧ i↵ r ⇤P ⇡ 2 .
• Given a formula '(x0 , . . . , xn 1 ) and P–names ⌧ 0 , . . . , ⌧ n 1 ,
p
i↵ for all q P p, q
⇤
0
n 1
)
P ¬'(⌧ , . . . , ⌧
1⇤P '(⌧ 0 , . . . , ⌧ n 1 ).
56
D. ASPERÓ
• Given formulas '0 (x0 , . . . , xn 1 ) and '1 (x0 , . . . , xn 1 ) and P–
names ⌧ 0 , . . . , ⌧ n 1 ,
p
⇤
P
'(⌧ 0 , . . . , ⌧ n 1 ) ^ '(⌧ 0 , . . . , ⌧ n 1 )
i↵ p ⇤P '0 (⌧ 0 , . . . , ⌧ n 1 ) and p ⇤P '1 (⌧ 0 , . . . , ⌧ n 1 ).
• Given a formula '(x, x0 , . . . , xn 1 ) and P–names ⌧ 0 , . . . , ⌧ n 1 ,
p
⇤
P
(8x)('(⌧ 0 , . . . , ⌧ n 1 ))
i↵ for every P–name ⌧ , p
⇤
P
'(⌧, ⌧ 0 , . . . , ⌧ n 1 ).
Note that there is no circularity in the definition of ⇤P for atomic
formula as it is by recursion on the rank of the corresponding names.
And neither it is for non-atomic formulas since the definition is by
recursion on their complexity.
Lemma 9.8. Let '(x0 , . . . , xn 1 ) be a formula, let ⌧ 0 , . . . , ⌧ n 1 be P–
names, and let p 2 P. Then:
(1) If p ⇤P '(⌧ 0 , . . . , ⌧ n 1 ) and q P p, then q ⇤P '(⌧ 0 , . . . , ⌧ n 1 ).
(2) p ⇤P '(⌧ 0 , . . . , ⌧ n 1 ) if and only if {q 2 P : q ⇤P '(⌧ 0 , . . . , ⌧ n 1 )}
is dense below p.
(3) If p 1⇤P '(⌧ 0 , . . . , ⌧ n 1 ), then there is some q P p such that
q ⇤P ¬'(⌧ 0 , . . . , ⌧ n 1 ).
[Proof: Exercise.]
Theorem 9.9. (Forcing Theorem) Suppose M is a transitive model of
(a fragment of ) ZF, P 2 M is a forcing notion, and for every p 2 P
there is a P–generic filter G over M with p 2 G. Let G be a P–generic
filter over M , '(x0 , . . . , xn 1 ) a formula, and let ⌧ 0 , . . . , ⌧ n 1 be P–
names in M . Then the following are equivalent.
(1) M [G] |= '(⌧G0 , . . . , ⌧Gn 1 )
(2) There is some p 2 G such that M |= p ⇤P '(⌧ 0 , . . . , ⌧ n 1 ).
Proof. By induction on the rank of names for atomic formulas and on
the complexity of '(x0 , . . . , xn 1 ) for non-atomic '(x0 , . . . , xn 1 ).
2 ⌧ : (=)). If G 2 ⌧G , choose (p, ⇡) 2 ⌧ such that G = ⇡G and
p 2 G. By induction we can choose q 2 G such that q ⇤P = ⇡. By
the lemma, {r P q : r ⇤P = ⇡} is dense below q. Then q ⇤P 2 ⌧
by definition of “q ⇤P 2 ⌧ ”.
((=) Suppose p 2 G and
D = {r 2 P : r
⇤
P
= ⇡ for some (q, ⇡) 2 ⌧, r P q}
is dense below p. Then by genericity we may choose r 2 G \ D and
(q, ⇡) witnessing r 2 D. By induction, G = ⇡G . But then q 2 G, and
MAGIC Set Theory lecture notes (Spring 2014)
therefore
G
57
= ⇡G 2 ⌧G (by definition of ⌧G ).
= ⌧ : (=)) Suppose G = ⌧G . Let D be the set of p 2 P such
that p ⇤P = ⌧ or for some (r, ⇡) 2 [ ⌧ , p P r and p ⇤P ⇡ 2
/
or p ⇤P ⇡ 2
/ ⌧ . Then D is dense, by the definiton of ⇤P = ⌧ and
the previous lemma. Hence there is some q 2 G \ D, and necessarily
q ⇤P = ⌧ (using the induction hypothesis).
((=) Suppose p 2 G and p ⇤P
= ⌧ . By induction hypothesis
together with the definition of p ⇤P = ⌧ , for all (r, ⇡) 2 [ ⌧ such
that r 2 G we have that ⇡G 2 G if and only if ⇡G 2 ⌧G . Hence G = ⌧G .
¬'(⌧ 0 , . . . , ⌧ n 1 ): (=)) Consider the set
D = {p 2 P : p
⇤
P
'(⌧ 0 , . . . , ⌧ n 1 ) or p
⇤
P
¬'(⌧ 0 , . . . , ⌧ n 1 )}
D 2 M is dense by definition of p ⇤P ¬'(⌧ 0 , . . . , ⌧ n 1 ). Hence there
is some p 2 G \ D by density. If p ⇤P '(⌧ 0 , . . . , ⌧ n 1 ), then M [G] |=
'(⌧G0 , . . . , ⌧Gn 1 ) by induction hypothesis. It follows that necessarily
p ⇤P ¬'(⌧ 0 , . . . , ⌧ n 1 ).
((=) Suppose p 2 G, M |= p ⇤P ¬'(⌧ 0 , . . . , ⌧ n 1 ), but M [G] |=
'(⌧G0 , . . . , ⌧Gn 1 ). By induction hypothesis there is p0 2 G such that
M |= p0 ⇤P '(⌧ 0 , . . . , ⌧ n 1 ). Let q P p, p0 , and note that M |= q ⇤P
'(⌧ 0 , . . . , ⌧ n 1 ) and M |= q ⇤P ¬'(⌧ 0 , . . . , ⌧ n 1 ) by the lemma. But
this contradicts the definition of q ⇤P ¬'(⌧ 0 , . . . , ⌧ n 1 ).
'0 (⌧ 0 , . . . , ⌧ n 1 ) ^ '1 (⌧ 0 , . . . , ⌧ n 1 ): For this case we note that
M [G] |= '0 (⌧G0 , . . . , ⌧Gn 1 ) ^ '1 (⌧G0 , . . . , ⌧Gn 1 )
i↵ M [G] |= '0 (⌧G0 , . . . , ⌧Gn 1 ) and M [G] |= '1 (⌧G0 , . . . , ⌧Gn 1 ) i↵ there
are p0 2 G and p1 2 G such that M |= p0 ⇤P '0 (⌧0 , . . . , ⌧ n 1 ) and
M |= p1 ⇤P '1 (⌧0 , . . . , ⌧ n 1 ) (by induction hypothesis) i↵ there is some
p 2 G such that M |= p ⇤P '0 (⌧ 0 , . . . , ⌧ n 1 ) ^ '1 (⌧ 0 , . . . , ⌧ n 1 ) ( ! is
true since p0 , p1 2 G and G is a filter, and
is true since p ⇤P '0 ^'1
⇤
⇤
implies p P '0 and p P '1 ).
(9x)('(x, x0 , . . . , xn 1 )): (=)) For this case suppose that
M [G] |= (9x)('(x, ⌧G0 , . . . , ⌧Gn 1 ))
and let ⌧ 2 M be a P–name such that M [G] |= '(⌧G , ⌧G0 , . . . , ⌧Gn 1 ).
By induction hypothesis we may fix some p 2 G such that M |= p ⇤P
'(⌧, ⌧ 0 , . . . , ⌧ n 1 ). But then
{q 2 P : M |= q
⇤
P
'( , ⌧ 0 , . . . , ⌧ n 1 ) for some P–name
is dense below p, and therefore M |= p
⇤
P
2 M} 2 M
(9x)('(x, ⌧ 0 , . . . , ⌧ n 1 )).
58
D. ASPERÓ
((=) The set D of p such that M |= p ⇤P '(⌧, ~⌧ ) for some P–name
⌧ 2 M or M |= p ⇤P ¬'(⌧, ~⌧ ) for all P–names ⌧ 2 M is dense and in
M . By genericity there is some p0 2 G \ D, and necessarily there is a
P–name ⌧ 2 M such that M |= p0 ⇤P '(⌧, ~⌧ ) as otherwise we reach a
contradiction as in previous case. But then M [G] |= '(⌧G , ⌧G0 , . . . , ⌧Gn 1 )
and hence M [G] |= (9x)('(⌧G0 , . . . , ⌧Gn 1 )).
⇤
Theorem 9.10. Let M be a transitive model of ZF, let P 2 M be a
forcing notion, and let G be P–generic over M . Then M [G] |= ZF. If,
in addition, M |= AC, then M [G] |= AC.
Proof. As M [G] is transitive and ! 2 M [G], M [G] satisfies Extensionality, Infinity, and of course Foundation.
Axiom of unordered pairs: Given P–names ⌧ 0 , ⌧ 1 , consider ⌧ =
{(1P , ⌧ 0 ), (1P , ⌧ 1 )}. Then ⌧G = {⌧G0 , ⌧G1 }.
Union: Given a P–name ⌧ , let
[
= {(p, ⇡) : p ⇤P ⇡ 2
⌧, rank(⇡) < rank(⌧ )}
By the forcing theorem,
[
⌧G \ {⇡G : ⇡ 2 M a P–name, rank(⇡) < rank(⌧ )}
G =
S
But any member of
⌧G is ⇡GSfor some name ⇡ 2 M such that
rank(⇡) < rank(⌧ ). Hence G = ⌧G .
Power set: Given a P–name ⌧ 2 M , let ↵ = max{rank(P), rank(⌧ )}
and = ↵ + !. Let
= {(p, ⇡) : p
⇤
P
⇡ ✓ ⌧, ⇡ 2 M, rank(⇡) < } 2 M
Then
G
= P(⌧G ) \ {⇡G : ⇡ 2 M a P–name, rank(⇡) < }
again by the forcing theorem. Now let ⇡ 2 M be a name such that
⇡G ✓ ⌧G . Let
⇡ ⇤ = {(p, x̃) : (p, ẋ) 2 ⇡, rank(x̃) < ↵, p
⇤
P
x̃ = ẋ}
⇤
Then rank(⇡ ⇤ ) < and it is easy to see that ⇡G
= ⇡G (since ⇡G ✓ ⌧G
and hence every member of ⇡G is x̃G for some x̃ with (q, x̃) 2 ⌧ for some
q, and therefore such that rank(x̃) < ↵). Hence, G = P(⌧G ) \ M [G],
so M [G] |= G = P(⌧G ).
Replacement: Let ⌧ 2 M be a name, and let f : ⌧G ! M [G] be
a function definable over M [G] possibly with parameters. Given an
ordinal ↵ 2 M , let
↵
= {(p, ⇡) : rank(⇡) < ↵, p
⇤
P
⇡ 2 range(f )}
MAGIC Set Theory lecture notes (Spring 2014)
59
(we are abusing notation here; f really refers to the definition of f ).
Then
↵
G
= range(f ) \ {⇡G : ⇡ 2 M a P–name, rank(⇡) < ↵}
once again by the forcing theorem. By the right instance of Replacement in M there is an ordinal ↵ 2 M such that if p 2 P, rank(⇡0 ) < ↵,
p ⇤P ⇡0 2 ⌧ , and p ⇤P f (⇡0 ) = ⇡, then there is such a ⇡ such that
rank(⇡) < ↵ (given such ⇡0 and p consider the minimal ↵ such that
there is a name ⇡ such that rank(⇡) = ↵ and p ⇤P f (⇡0 ) = ⇡). Now it
↵
is easy to check that G
= range(f ).
Finally suppose the Axiom of Choice holds in M . Let ⌧ 2 M be a
name and let < be a well–order on the set of names 2 M such that
rank( ) < rank(⌧ ).
Now we can well–order ⌧G as follows: Given x, y 2 ⌧G , x <G y i↵
ẋ < ẏ, where ẋ is the <–first name ż such that żG = x and ẏ is the
<–first name ż such that żG = y.
⇤
The following is a corollary of the above proof.
Corollary 9.11. Suppose F ✓ ZF is finite. Then there is a finite F 0 ✓
ZF such that for every transitive M |= F 0 , every forcing notion P 2 M ,
and every P–generic filter G over M , M [G] |= F . And similarly with
ZFC instead of ZF.
10. Two forcing constructions
The first of our constructions will be adding many Cohen reals to a
model of (a fragment) of ZFC. With this construction we will show the
consistency of the continuum being arbitrarily large.
10.1. Adding many Cohen reals.
Definition 10.1. Let ↵ be an ordinal and let Add(!, ↵) be the following forcing notion. A condition in Add(!, ↵) is a finite function
p ✓ (↵ ⇥ !) ⇥ 2. Given Add(!, ↵)–conditions p0 , p1 , p1 Add(!, ↵) p0 i↵
p0 ✓ p1 .
Lemma 10.2. Let M be a transitive model of (enough of ) ZF, let
↵ 2 M be an ordinal and let < ↵. If G is generic for Add(!, ↵)M
(= Add(!, ↵)) over M , then
cG = {(n, ✏) 2 ! ⇥ 2 : ((↵, n), ✏) 2 p for some p 2 G}
is a Cohen real over M (in other words, cG is 2<! –generic over M ).
Furthermore, if < 0 < ↵, then cG 6= cG0 .
60
D. ASPERÓ
Proof. Let D 2 M be a dense subset of 2<! . Then D̃ = {p 2
Add(!, ↵) : {(n, ✏) : ((↵, n), ✏) 2 p} 2 D} is obviously dense. Hence,
0
cG
< ↵, 0 6= .
↵ \ D 6= ;. Now suppose
Claim 10.3. The set
D
,
0
= {p 2 Add(!, ↵) : p( , n) 6= p( 0 , n) for some n 2 !} 2 M
is dense.
Proof. Obvious, since conditions in Add(!, ↵) are finite functions and
thus involve only finitely many n 2 !.
⇤
By the claim, there is some n < ! such that cG (n) 6= cG0 (n), and so
cG 6= cG0 .
⇤
Corollary 10.4. If M is a transitive model of enough of ZFC, ↵ 2 M
is an ordinal, and G is Add(!, ↵)–generic over M , then M [G] |= 2@0
|↵|.
In this corollary, if in particular ↵ = @1000 , then M [G] |= 2@0
Is this enough to conclude the consistency of ZFC+2@0 @1000
relative to that of ZFC? Not quite yet. For all we know so far it could
M
be that M [G] |= |@M
1000 | < @1000 . It could even be that @1000 has become
countable in M . We will see next that this does not happen, and in
M [G]
fact @M
1000 = @1000 .
|@V1000 |.
10.2. Chain condition and cardinal preservation.
Definition 10.5. Given a partial order P, A ✓ P is an antichain if p
and p0 are incompatible (i.e., there is no q 2 P, q P p, q P p0 ) for all
p, p0 2 A, p 6= p0 . A ✓ P is a maximal antichain if A is an antichain
and it is ✓–maximal with respect to ✓ (equivalently, for every p 2 P
there is some p0 2 A and some q 2 P such that q P p, p0 ).
Definition 10.6. Given a cardinal , a partial order P has the –chain
condition (P has the –c.c.) i↵ there are no antichains A of P such that
|A| = . If P has the @1 –c.c., then we also say that P has the countable
chain condition (P is c.c.c.).
Fact 10.7. (ZFC) Suppose P is a forcing notion and D ✓ P is dense.
Then there is a maximal antichain A of P such that A ✓ D.
Proof. Let (p⇠ )⇠<µ be an enumeration of D, where µ = |D|. Build
(⇠i )i<µ0 , for some cardinal µ0  µ, as follows:
• ⇠0 = 0
• Let i > 0 be given and suppose (⇠i0 )i0 <i defined.
MAGIC Set Theory lecture notes (Spring 2014)
61
– If there is no p 2 D such that p is incompatible with all
p0 2 {p⇠i0 : i0 < i}, then we let A = {p⇠i0 : i0 < i} and
stop.
– If there is some p 2 D such that p is incompatible with all
p0 2 {p⇠i0 : i0 < i}, then we let i be least such that p⇠i is
incompatible with all p⇠i0 .
A is an antichain of P by construction. Suppose it is not maximal.
Let p 2 P be such that p is incompatible with all p0 2 A. Since D is
dense, let q 2 D, q P p. By construction of A there is some p0 2 A
and some q 0 2 P such that q 0 P p0 , q. But then q 0 P p0 , p, and so p is
compatible with some p0 2 A. Contradiction.
⇤
Lemma 10.8. Suppose P is a forcing notion, <  are infinite cardinals, f˙ is a P–name, and p 2 P is a condition such that
p P f˙ is a function, f˙ : ˇ ! ̌
Suppose P is c.c.c. Then p P f˙ is not onto ̌.
Proof. For every ⇠ 2
}.
let D⇠ = {p 2 P : p
P
ˇ =↵
f˙(⇠)
ˇ for some ↵ 2
Claim 10.9. D⇠ is dense.
Proof. Given p 2 P, let G be P–generic, p 2 G. Since p P f˙ : ˇ !
̌ is a function, f˙G :
!  is a function. Let ↵ <  be such that
f˙G (⇠) = ↵. By the forcing theorem there is some p0 2 G such that
ˇ =↵
p0 P f˙(⇠)
ˇ . Since p, p0 2 G, we may find q 2 P, q  p, p0 . But
ˇ =↵
then q P f˙(⇠)
ˇ.
⇤
Given ⇠ < , let A⇠ ✓ D⇠ be a maximal antichain in P. By our
hypothesis, |A⇠ |  @0 . Given ⇠ < , let
A⇠ = {↵ <  : p
P
ṗ(⇠) = ↵ for some p 2 D⇠ }
Claim 10.10. |A⇠ |  @0 .
Proof. For every ↵ 2 A⇠ let q↵ 2 A⇠ compatible with some condition
ˇ =↵
p↵ 2 P forcing f˙(⇠)
ˇ . Since p↵ and q↵ are compatible and q↵ P
0
0
˙
f (⇠) = ↵ for some ↵ , it must be that ↵0 = ↵. Hence A⇠ ✓ {↵ : p P
ˇ = ↵ for some p 2 A⇠ }. But |{↵ : p P f˙(⇠)
ˇ = ↵ for some p 2
f˙(⇠)
A⇠ }|  @0 .
⇤
S
Finally, by the forcing theorem, range(f˙G ) ✓ ⇠< A⇠ for every P–
S
generic G over M , p 2 G. But M |= | ⇠< A⇠ |  | ⇥ @0 | = < 
62
D. ASPERÓ
S
implies ⇠< A⇠ 6=  (it is easy to see that |! ⇥ | = for every infinite
cardinal [Exercise.]). Hence f˙G cannot be onto  for any such G. ⇤
Corollary 10.11. Suppose M is a transitive model of enough of ZFC,
P 2 M and M |= P is c.c.c. Suppose G is P–generic over M . Then, for
every  2 M such that M |=  is a cardinal, M [G] |=  is a cardinal.
M [G]
In particular, for every ordinal ↵ 2 M , @M
.
↵ = @↵
Lemma 10.12. (ZFC) For every ordinal ↵, Add(!, ↵) has the countable chain condition.
Proof. Suppose, towards a contradiction, that (p⌫ )⌫<!1 enumerates an
uncountable antichain of Add(!, ↵). Since {dom(p⌫ ) : ⌫ < !1 } is a
collection of finite sets, by the –system lemma we know that there is
X ✓ !1 , |X| = @1 , such that {dom(p⌫ ) : ⌫ 2 X} forms a –system
with root R; in other words, for all distinct ⌫, ⌫ 0 in X, dom(p⌫ ) \
dom(p⌫ 0 ) = R. Note that there are only finitely many functions from
the finite set R to 2 (in fact there are exactly 2|R| such functions). Since
|X| = @1 and 2|R| is finite, it follows that there are distinct ⌫, ⌫ 0 2 X
such that p⌫ R = p⌫ 0 R. But then,
p⌫ [ p⌫ 0 : dom(p⌫ ) [ dom(p⌫ 0 ) ! 2
is a function. Since q := p⌫ [ p⌫ 0 extends both p⌫ and p⌫ 0 , we have that
q Add(!, ↵) p⌫ , p⌫ 0 . A contradiction.
⇤
Corollary 10.13. Suppose M is a transitive model of enough of ZFC,
 2 M is such that M |= “ is a cardinal”, and G is an Add(!, )–
generic filter over M . Then
(1) for every ordinal 2 M , M |= “ is a cardinal” if and only if
M [G] |= “ is a cardinal”, and
(2) M [G] |= 2@0 
@0
In particular, if ↵ 2 Ord \M and  = @M
@↵ .
↵ , then M [G] |= 2
10.3. –closure and not adding new reals. We see a companion
result next: The consistency of ZFC + CH, relative to the consistency
of ZFC, using forcing (the original proof of this result is due to Gödel,
long before forcing was invented: he built the constructible universe,
L, and proved that L |= ZFC + CH. We don’t have time here to say
much about L except for its definition, which we have already seen (cf.
after Corollary 7.10).
Definition 10.14. A partial order P is –closed i↵ for every P –
decreasing sequence (pn )n<! of P–conditions there is some q 2 P such
that q P pn for all n.
MAGIC Set Theory lecture notes (Spring 2014)
63
Lemma 10.15. Suppose M is a transitive model of enough of ZFC,
P 2 M is a forcing notion, M |= P is –closed, G is P–generic over
M , and f : ! ! Ord, f 2 M [G]. Then f 2 M .
Proof. For every n < !, Dn = {p 2 P : p P f˙(n) = ↵ for some ↵} 2
M is dense. Also, of course (Dn )n<! 2 M . It suffices to see that there
is an ordinal ✓ 2 M such that
D = {p 2 P : p P f˙ = ǧ for some function g 2 V M } 2 M
✓
is dense.
For this let p 2 P. Build, in M , a P –decreasing sequence (pn )n<!
of conditions in P extending p and such that pn+1 2 Dn for all n.
Let p0 = p. Given pn , we can find pn+1 since Dn is dense. Finally,
let q P pn for all n. q exists since P is –closed in M . But then
q 2 D.
⇤
Definition 10.16. Let Coll(P(!), !1 ) be the following partial order:
p 2 Coll(P(!), !1 ) i↵ p ✓ !1 ⇥ P(!) is a function such that |p|  @0 .
Given Coll(P(!), !1 )–conditions p0 , p1 , p1 Coll(P(!),!1 ) p0 i↵ p0 ✓ p1 .
Lemma 10.17. Coll(P(!), !1 ) is –closed.
Proof. Obvious, since the union of a decreasing !–sequence in the partial order Coll(P(!), !1 ) is a countable function.
⇤
Lemma 10.18. For every transitive model M of enough of ZFC and
every r ✓ !, r 2 M , Dr = {p 2 Coll(P(!), !1 )M : r 2 range(p)} 2 M
is a dense subset of Coll(P(!), !1 )M .
Proof. If p 2 Coll(P(!), !1 )M and r 2
/ range(p), then p [ {(↵, r)} 2 D,
where ↵ is any ordinal in !1M \ dom(p) 6= ;.
⇤
Similarly we can prove:
Lemma 10.19. For every transitive model M of enough of ZFC and
every ↵ 2 !1M , E↵ = {p 2 Coll(P(!), !1 )M : ↵ 2 dom(p)} 2 M is a
dense subset of Coll(P(!), !1 )M .
Corollary 10.20. Let M be a transitive model of enough of ZFC and
let G be a generic filter for Coll(P(!), !1 )M . Then M [G] |= 2@0 = @1 .
S
Proof. By the last two lemmas above, G : !1M ! P(!)M is a surjection. Also, by the previous two lemmas, P(!)M [G] = P(!)M since
Coll(P(!), !1M ) is –closed in M and therefore does
S not add new functions f : ! ! 2. But then M [G] thinks that G : !1 ! P(!) is
onto all of P(!).
⇤
64
D. ASPERÓ
By the above corollaries, together with the metamathematical considerations at the beginning of this section on forcing, we have proved
the following:
• Con(ZFC) implies Con(ZFC + CH).
• Con(ZFC) implies Con(ZFC +2@0
@2 ), Con(ZFC +2@0
@3 ),
@0
@0
Con(ZFC +2
@2454536254465 ), Con(ZFC +2
@@@785665 ),
etc.
It is not difficult to prove, by the same forcing construction we have
seen (adding many Cohen reals) using a slightly more refined analysis,
that 2@0 can consistently be exactly @2 , @3 , @2454536254465 , @@@785665 , etc.
It is also easy to change the value of 2@1 , 2@2 , 2@!+1 , and so on,
by forcing. And, in fact, any constellation of values is possible for
cardinals as the above (and many others), subject to some very simple
rules. For example, it is consistent – modulo Con(ZFC) of course –
that 2@0 = @1 + 2@1 = @27 + 2@2 = 2@3 = @28 + 2@!+1 = @!1 , etc.
On the other hand, there are very deep and surprising cardinal arithmetical facts one can prove in ZFC for other cardinals! For example:
Theorem 10.21. (Shelah) (ZFC) Suppose 2@n < @! for all n < !.
Then 2@! < @!4 .
A big open question in “pcf theory” is whether the above bound @!4
can be improved to something like @!1 .
Forcing has applications not only in cardinal arithmetical. In fact it
can be used to build models of set theory with many di↵erent properties.
One example (J. Moore): If M is a transitive countable model of (a
fragment of) ZFC +LC, where LC is some (relatively weak) “large
cardinal axiom”, then there is a forcing extension M [G] such that
M [G] thinks that there is a basis of exactly 5 elements for the uncountable linear orders; in other orders, in M [G] it holds that there is
B = {L0 , L1 , L2 , L3 , L4 } such that each Li is an uncountable linear order, and such that if L is any uncountable linear order, then L contains
an isomorphic copy of at least one Li . In this model, 2@0 > @1 . In fact,
2@0 = @1 implies that if B is a basis for the uncountable linear orders,
then |B| = 2@1 .
11. Improving ZFC?
We have seen that ZFC does not decide the value of 2@0 (i.e., does not
decide the ordinal ↵ such that 2@0 = @↵ ). Also, ZFC does not decide
whether or not there is a 5–element basis for the uncountable linear
MAGIC Set Theory lecture notes (Spring 2014)
65
orders, as well as infinitely many other questions in mathematics (hundreds or thousands of these questions have been explicitly considered
by mathematicians). Given the independence phenomenon, should we
gather these questions as meaningless?
Not necessarily. We can hope to find natural axioms which, when
added to ZFC, decide these questions. The most successful and so far
best understood family of such axioms come under the name of “large
cardinal axioms”. The weakest such axiom asserts that there is an
uncountable cardinal  such that
•  is a limit cardinal, and
•  is regular (i.e.,  is not the union of less than –many sets of
size less than ).
Such a  is called a (strongly) inaccessible cardinal.
It is not possible to prove
Con(ZFC) =) Con(ZFC +“There is an inaccessible cardinal”)
because ZFC +“There is an inaccessible cardinal” proves Con(ZFC) (unless we can prove ¬ Con(ZFC), that is): The reason is that if  is inaccessible, then V |= ZFC. Now suppose towards a contradiction that
we can prove the arithmetical statement
Con(ZFC) =) Con(ZFC +“There is an inaccessible cardinal”)
in, say, Peano Arithmetic or ZFC. But
ZFC +“There is an inaccessible cardinal”
proves Con(ZFC). Hence
ZFC +“There is an inaccessible cardinal”
proves Con(ZFC +“There is an inaccessible cardinal”), so
ZFC +“There is an inaccessible cardinal”
is inconsistent by Gödel’s 2nd Incompleteness Theorem. Hence ZFC
would be inconsistent.
We say that ZFC + “There is an inaccessible cardinal” is strictly
stronger than ZFC.
Similarly:
• ZFC + “There is a weakly compact cardinal” is strictly stronger
than ZFC + “There is an inaccessible cardinal”.
• ZFC + “There is a measurable cardinal” is strictly stronger
than ZFC + “There is a weakly compact cardinal”.
66
D. ASPERÓ
• ZFC + “There is a Woodin cardinal” is strictly stronger than
ZFC + “There is a measurable cardinal”.
• ZFC + “There is a supercompact cardinal” is strictly stronger
than ZFC + “There is a Woodin cardinal”.
• ZFC + “There is a huge cardinal” is strictly stronger than ZFC
+ “There is a supercompact cardinal”.
• ...
The theories of the form ZF + LC (or ZFC+LC), where LC is a large
cardinal axiom, form a hierarchy of theories natural linearly ordered
by consistency strength.
It is a remarkable empirical fact that all ‘natural’ theories arising
in mathematics can be interpreted relative to some such theory, and
that in many cases they can be proved to be equiconsistent with such
a theory. In this sense, large cardinal axioms provide a very natural
template for extending ZF or ZFC.
Sufficiently strong large cardinal axioms have also remarkable “astrological” properties: They prove things about ‘simply definable’ sets of
reals that ZFC does not decide, if consistent. It is surprising that large
cardinal axioms, that assert the existence of certain objects very high
in the cumulative hierarchy with properties that can only be recognised
by looking very high, have direct influence on objects lying very low
in the hierarchy. For example: If there are infinitely many Woodin
cardinals, then every projective set of reals has the usual regularity
properties (it is Lebesgue measurable, has the Baire property, and has
the perfect set property). On the other hand, in the absence of such
large cardinals it is possible that there be a closed set X ✓ R3 such
that Y is not Lebesgue measurable, where Y is the projection to R
of the complement, in R2 , of the projection of X to R2 . This Y is,
by construction, a projective set, in fact quite low in the projective
hierarchy.
(Un)fortunately one can prove that large cardinal axioms do not
decide 2@0 . For this one has to look at other families of axioms, compatible with large cardinal axioms (for example “forcing axioms”).
Acknowledgements: Thanks to Daoud Siniora for a list of corrections.
References
[1] T. Jech, Set Theory: The Third Millenium Edition, Revised and Expanded,
Springer, Berlin (2002).
MAGIC Set Theory lecture notes (Spring 2014)
67
[2] K. Kunen, Set Theory, An introduction to independence proofs, North-Holland
Publishing Company, Amsterdam (1980).
[3] E. Mendelson, Introduction to mathematical logic, 5th. ed., 2009.
David Asperó, School of Mathematics, University of East Anglia,
Norwich NR4 7TJ, UK
E-mail address: [email protected]