Set Theory, Forcing and Real Line
Giorgio Laguzzi
March 21, 2013
Abstract
We give a very brief survey on ZFC theory (Zermelo-Fraenkel Set Theory) and we present an intuitive introduction to the method of forcing and
some applications to the real line. Our exposition will be very informal,
without any claim of completeness and rigour. The idea is just to give a
very intuitive idea of what Set Theory and forcing-method are, and why
they are interesting and useful from the viewpoint of Mathematics. In
the final part we also expose some results recently obtained, concerning
regularity properties of the real numbers, such as Lebesgue measurability,
Baire property, Ramsey property and Silver measurability.
Axioms of ZFC
First of all, we would like to point out that the present paper really consists of a
very informal introduction to Set Theory, and it is intended to just give a gentle
and not too pedant presentation. For a very detailed and technical introduction
to ZFC and forcing method, we refer the reader to [3], which we consider the
most rigorous book on this topic. On the contrary, for a less demanding presentation, though still very precise and complete, one can see [6] and [1].
ZFC is a first order logic theory with equality and a unique binary relation
symbol ∈ (intuitively thought as “membership”).
1. Set existence: ∃x(x = x).
2. Extensionality: ∀x∀y(∀z(z ∈ x ↔ z ∈ y) → x = y).
3. Comprehension Scheme: For every formula ϕ with free variables among
x, z, v1 , . . . , vn ,
∀z∀v1 . . . vn ∃y∀x(x ∈ y ↔ x ∈ z ∧ ϕ)
4. Foundation: ∀x[∃y(y ∈ x) → ∃y(y ∈ x ∧ ¬∃z(z ∈ x ∧ z ∈ y))].
5. Pairing: ∀x∀y∃z(x ∈ z ∧ y ∈ z).
6. Union: ∀F∃A∀Y ∀x(x ∈ Y ∧ Y ∈ F → x ∈ A).
7. Replacement Scheme: For every formula ϕ with free variables among
x, y, A, v1 , . . . , vn ,
∀A∀v1 . . . vn (∀x ∈ A∃!yϕ → ∃Y ∀x ∈ A∃y ∈ Y ϕ).
1
On the basis of Axiom 1-7 one can define ⊆(subset), ∅ (empty set), S (ordinal successor, i.e., S(x) = x∪{x}), and the notion of well-ordering (the latter being an antisymmetric, reflexive, transitive binary relation R on a set
X such that: ∀x, y ∈ X(Rxy∨Ryx) and ∀A ⊆ X∃x ∈ A(x is R-minimal)).
We can then complete our list of axioms as follows:
8. Infinity: ∃x(0 ∈ x ∧ ∀y ∈ x(S(y) ∈ x)).
9. Power Set: ∀x∃y∀z(z ⊆ x → z ∈ y).
10. Choice: ∀A∃R(R well-oders A).
The intuitive meaning of the Comprehension Scheme is that given a set z
and a formula ϕ, one can define a set y ⊆ z of those elements in z satisfying
ϕ. Note that it does not mean that given any formula, one can define a set
consisting of those x satisfying such a formula. As a counterexample, one may
consider the formula ϕ(x) ≡ x = x. In this case, we get the class
V = {x : x = x}.
(In this expression the symbol “=” has to be meant in two different ways; the
former occurs as an equality in the metatheory, the latter inside the theory. This
kind of confusion between symbols of the theory and symbols of the metatheory
will come further, and one should keep in mind this difference). Obviously V
cannot be a set itself, for otherwise we would have V ∈ V, contradicting the
axiom of foundation. Such kind of object is called proper class, i.e., a class
which is not a set. Any proper class can be described in ZFC, just via the
corresponding formula, but it is not an object of the theory; that means the
variables run only over sets, and not over proper classes.
Some popular notions from the viewpoint of ZFC
Ley us now try to become familiar with some common objects definable from
ZFC.
• Natural numbers are identified with sets recursively constructed starting
from the empty set ∅:
0=∅
1 = {∅} (note: 1 = ∅ ∪ {∅} = S(0))
n + 1 = n ∪ {n}.
• The set of all natural numbers is denoted by ω; note that ω can be even
viewed as the union of all natural numbers. Since the successor operator
S can be applied to any set, we can then define S(ω) = ω ∪ {ω}, which is
also denoted by ω + 1, and we can then recursively define
ω + 2 = S(ω + 1) = ω ∪ {ω} ∪ {ω + 1}.
ω + n + 1 = S(ω + n) = ω ∪ {ω} ∪ {ω + 1} · · · ∪ {ω + n}.
S
ω + ω = {ω + n : n ∈ ω}. (ω + ω is denoted by ω · 2 as well.)
we can keep on such a construction
in order to define the ω · 2 + n’s
S
and then their limit ω · 3 = {ω · 2 + n : n ∈ ω}.
2
one can recursively get ω · n and then ω · ω =
is denoted by ω 2 .)
S
{ω · n : n ∈ ω} (ω · ω
...
Remark 1. Note that 0, 1, . . . , n, . . . and ω + 1, ω + n, . . . , ω · 2, ω · ω can all be
defined by recursion theorem. The unique object previously mentioned having a
different flavour is just ω. In fact, for instance, if we look at ω · 2, we realize that
it is defined by using elements which are “previously constructed” (the ω + n’s)
plus replacement and union (running over ω, which we assume to be “already”
defined):
[
ω · 2 = {ω + n : n ∈ ω}.
On the contrary, for ω the situation is quite different; as limit, we would like to
write it as a union, i.e.,
ω = {n : n ∈ ω}!!!
But this is rather senseless, for the union defining ω runs over ω itself, which we
want to define . . . This is somehow a symptom of the fact that we really need
the axiom of infinity to ensure the existence of such a set.
Note that all the sets α just defined are countable. i.e., there is a bijection
j : α → ω. A natural question which arises is whether in ZFC one can prove
the existence of uncountable sets, i.e., sets that are not in bijection with ω. The
answer is positive; in fact, let us define
U = {α : ∃j : α → ω bijection}
and then let ω + = {α : α ∈ U}. It is clear that ω + is uncountable, for if it
were not then ω + would be in U and so ω + ∈ ω + , contradicting the axiom of
foundation. This sets is usually denoted by ω1 .
In a similar fashion one can then define ω2 , ω3 , . . . , ωω and so on.
S
Remark 2. All these sets that we have introduced have the particular feature of
being transitive (i.e., every element of the set is also a subset) and well-ordered
by ∈. These sets are called ordinals. When dealing with ordinals, we can easily
define an ordering in the following way: α < β iff α ∈ β.
Another important notion used in Set Theory is that of cardinal : an ordinal
α is a cardinal iff there is no β < α which is in bijection with α itself. So for
example, ω, ω1 , ωn are cardinals, whereas ω + ω, ω · ω are ordinals, but not
cardinals.
Finally, we introduce the notion of regular cardinal:
• given two ordinals α and β, we say that α is cofinal in β iff there is a
function f : α → β such that for every γ1 ∈ β there is γ0 ∈ α such that
f (γ0 ) > γ1 .
• a cardinal κ is said to be regular iff there is no λ < κ such that λ is cofinal
in κ.
Examples of regular cardinals are the ωn ’s, with n ∈ ω, whereas a counterexample is ωω , since the function f : ω → ωω , such that f (n) = ωn , is cofinal.
Non-regular cardinals are called singular.
3
The class of ordinal numbers is denoted by ON; more precisely,
ON = {x : x is an ordinal}.
Note that ON is transitive and well-ordered by ∈. Nevertheless, ON is not an
ordinal, since it is not a set, for otherwise we would have ON ∈ ON. Analogously,
the class of all cardinals CN is proper.
Models of ZFC
In the previous section, we introduced some popular notions and we also show
how one can recover some elementary notions (like natural numbers) under
the viewpoint of Set Theory. It is not hard to show that actually many other
common notions can fall into the range of action of this theory, such as those
of function, real number, Boolean algebra, group, and so on. Even more, one
can show that ZFC is so powerful that it can define the notion of structure,
model and the language of ZFC too, so that it can somehow study itself. This
self-reference could sound very suspicious, at first.
Further, note that in the previous section we somehow cheat; when we said
“ from the viewpoint of ZFC ”, what we actually meant?
Formally ZFC is nothing more than a theory expressed in the first order language. In the previous section, the meaning “ from the viewpoint of ZFC ” must
be interpreted as “ from the viewpoint of the intuitive interpretation of ZFC ”,
i.e., the natural structure where we interpret the symbol ∈ as membership and
we think of variables x, y, z as ranging over sets. The intuitive model that we
had (and we have) in mind can be presented in the following way. Consider the
recursive construction
N0 = ∅
Nα+1 = P (Nα ) (i.e., power set of Nα )
S
Nλ = λ = {Nα : α < λ}.
S
Finally put N = {Nα : α ∈ ON}. Such formal definition of N provides the
model that we intuitively had in mind.
Nevertheless, an interpretation of our language with ∈ could be completely
different. For instance, if we consider the structure having as universe the real
line R and we interpret ∈ as < then we get a legal interpretation of our language.
Obviously, one can immediately note that not all the axioms of ZFC are satisfied
by this structure; for instance, the statement “ ∃x(x = ∅) ” can be easily derived
from ZFC, but it is not true in the structure (R, <), since the latter satisfies the
statement “ ∀x∃y(y ∈ x) ”.
The structure we just mentioned is not a model of ZFC and so one can
object that it is not interesting for our purpose. Nevertheless, our intention was
just to show how our language can be seen under a completely different (but
still admissible) light. With the first attempt we have been unfortunate. So the
question which arises is to find different structures which are models of ZFC.
But, what we mean with the word “model”? Here, one has to be careful,
since there are two formally different ways to define such a notion in ZFC.
1 - Relativization. let M be any class and E be a binary class-relation
(intuitively, M is the universe and E is the interpretation of ∈). For every
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formula φ the relativization φM,E is recursively defined (on the complexity of φ)
as follows:
• (x = y)M,E is x = y, and (x ∈ y)M,E is E(x, y);
• (ψ ∧ χ)M,E is ψ M,E ∧ χM,E ;
• ∀xψ is ∀x(M(x) ⇒ ψ M,E ).
This definition means that, given some formula φ, one can produce a formula
φM,E which involves both the formula with one free variable defining the class M,
say θM (x, . . . ) (where . . . represent possible parameters), and the formula with
two free variables defining the class E, say θE (x, y, . . . ). Hence, one formally
obtains a formula φM,E expressed in the first order language of ZFC, and one
can therefore, for instance, ask whether or not ZFC ` φM,E . So, given a set of
sentences S, one says that hM, Ei is a model for S iff for every φ ∈ S, one has
ZFC ` φM,E .
2 - Model Theory. Another way to express the notion of model is by
formalizing within ZFC all the notions of Model Theory. For instance, one can
assign to each sentence φ a natural number nφ via the so called Gödelization.
Further, one can also formalize the notion of set-structure A = hA, Ei and the
∗
satisfactory relation, say |= . In such a way, we can obtain the formula
θ(A, E, nφ ) ≡ hA, Ei |=∗ nφ .
These two methods are formally very different. Nevertheless, a fairly standard proof on the complexity of a sentence φ shows the following metatheorem.
Theorem 3. Let φ be a formula in the language of ZFC. Then
ZF ` ∀hM, Ei(φM,E ⇔ θ(M, E, nφ )).
When M , E are proper classes, we cannot use the universal quantification
in the last formula. Nevertheless, we could get something similar, by fixing the
two formulae θM and θE representing them.
Theorem 4. Let φ be a formula in the language of ZFC, and fix two proper
classes M and E described via the formula θM and θE , respectively. Then
ZF ` φM,E ⇔ θM,E (nφ ).
Note that, in the right side, the formula θM,E (nφ ) denotes a kind of reformulation of θ(M, E, nφ ), since M, E cannot be parameters any longer.
By virtue of such (meta)theorems, from now on, we will adopt a sort of
compromise, by using the notation hM, Ei |= φ to denote φM,E (we omit E
when this is meant as the standard interpretation of membership). Throughout
this paper, all of the occurrences of statements like hM, Ei |= φ will be meant
in the sense of the relativization.
Let us now make some interesting and basic remarks.
• By Gödel’s incompleteness theorem, there are statement φ such that neither φ nor ¬φ are provable in ZFC. So in this case, if we pick M |= ZFC+φ
and N |= ZFC + ¬φ, we get M and N not being isomorphic. Except for
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the usual artificial statement introduced by Gödel, many other natural
examples have come out along the years (natural w.r.t. the intuitive interpratation of ZFC). Among these, let us consider the popular Continuum
Hypothesis CH, asserting that the set of subsets of ω has cardinality ω1 ,
briefly 2ω = ω1 . Gödel proved that the universe of constructible sets L
satisfies CH, whereas Cohen in the 1960s invented the method of forcing
to build a model for ZFC+¬CH. Both L and the method of forcing will
be briefly introduced in the coming pages.
• In some sense one can show that in ZFC there is a countable model of ZFC
itself. Firstly, what does model mean here?
By Gödel’s second incompleteness theorem, ZFC cannot prove its own
consistency, and so it cannot formally show that there is a model of itself.
More precisely,
ZFC 6` “ ∃M |= ZF C ”,
Hence the meaning of the statement “ZFC proves there is a countable
model of ZFC” should be differently meant. In fact, the real meaning will
be: given a finite list φ0 , . . . , φn of axioms of ZFC, then
ZFC ` ∃M(M |= φ0 ∧ · · · ∧ φn ).
This result is known as Reflection Theorem. Moreover, by Mostovsky’s
Collapse, one can make such a model being transitive.
Secondly, what does countable mean? In the previous section we have
shown that in ZFC one can prove that there are ordinals which are uncountable, like ω1 . Hence, one would say that any transitive model of ZFC
has to contain ω1 as an element and so also any of its element; but since
ω1 is uncountable we should have uncountably many such elements inside
the model!
What’s the trick? One can actually recognize two inaccuracies in the above
argument, which are somehow the two sides of the same coin. Firstly,
we implicitly assumed the interpretation of ∈ being as “membership”,
which is obviously not the unique possibility. In fact, as we mentioned
at the beginning of this section, the interpretation of the binary relation
symbol ∈ may vary, so that its meaning can be faraway from the intuitive
one of “membership”. Secondly, the ω1 that we are picking is the first
uncountable cardinal from the viewpoint of M, but it could be anything
else from an external viewpoint.
Looking at that from a different viewpoint, this amazing fact does not
seem so . . . amazing. In fact, if one thinks of the usual construction to
prove Gödel’s completeness theorem, it turns out that such a model has
the same cardinality of the language in which the theory is expressed,
therefore it is countable for ZFC. The intuitive reason for which such
Gödel’s model can be faraway from a standard natural interpretation is
mainly because it is somehow built on the syntax.
Note that, if we work inside a stronger theory than ZFC, we could really get a
set model of ZFC. For example, let us assume the statement
I ≡ ∃κ > ω(κ is a cardinal ∧ ∀α < κ(2α < κ))
6
(such kind of cardinal is called inaccessible.) One may then show that
ZFC+I ` “ Nκ |= ZFC ”.
Nevertheless, we are not interested in this kind of strong results. In fact, the
aim of Set Theory is to study the consistency of a certain theory T relatively
to some other theory T 0 ; or in other words, assuming T being consistent, what
can one say on the consistency of T 0 ?
Obviously, the way to really build inside T a model M for T 0 can be done
only when T 0 is strictly weaker than T , because of Gödel’s second incompleteness
theorem. It is then useless for our question, since the consistency of a weaker
theory (T 0 ) is simply implied by the consistency of the stronger one (T ).
Hence, our question requires a different method. The following well-known
fact sheds light on such results of relative consistency.
Lemma 5. Let T and T 0 be two sets of sentences. Assume that in T one can
prove, for some class M, for every φ ∈ T 0 , M |= φ. Then
Consistency of T → Consistency of T 0 .
The latter implication is usually denoted with Con(T ) → Con(T 0 ). The idea
is simply that if one can derive a contradiction from T 0 , then some contradiction
could also be derived in T . The idea of the proof is very simple; assume we can
derive a contradiction from T 0 , i.e, some formula of the form φ ∧ ¬φ. Hence, by
our assumption, we also get M |= φ ∧ ¬φ, indeed M |= φ and M |= ¬φ. But that
means
T ` “ M |= φ and M 6|= φ ”.
which is a contradiction.
Let us conclude this section with two well-known examples of relative consistency results and with a mysterious remark.
Example 6. We briefly introduce Gödel’s constructible universe L. Firstly,
remind that a set y is said to be definable over a structure A iff there exists a
formula ϕ(x) in the language of A such that
z ∈ y iff A |= ϕ(z).
For any set x, put Def(x) = {y ⊆ x : y is definable overhx, ∈i}. We then define
L by recursion as follows:
L0 = ∅
Lα+1 = Def(Lα )
S
Lλ = α<λ Lα .
S
We finally define the constructible universe as L = α∈ON Lα . Gödel proved
that such a structure is a model for ZFC, and it satisfies CH as well.
It is not hard to show that L can be defined inside ZF, without any need of
AC. Moreover, one can also check that for any φ ∈ ZFC, we have ZF ` “ L |=
φ ”. Hence, the following consistency result holds:
Con(ZF) → Con(ZFC + CH)
7
Example 7. One can show that Von Neumann’s hierarchy, previously introduced, can be defined inside ZF− , where the latter is the theory of ZermeloFraenkel without axiom of choice and foundation. Rather amazingly, one can
show that the axiom of foundation is however satisfied by N, and so
Con(ZF− ) → Con(ZF).
Remark 8. We would like to show a strange thing turning out from our results.
Let {φn : n ∈ ω} be the list of axioms of ZFC. Moreover, put Fn = φ0 ∧ φ1 ∧
· · · ∧ φn , for every n. By the reflection theorem we mentioned above, it follows
that, for every natural number n,
ZFC ` “∃M |= Fn ”.
So we would say ZFC ` “∀n ∈ ω∃M |= Fn ” and then, by compactness, we get
ZFC ` “∃M |= ZFC ”.
Contradiction . . . What went wrong?
Real Line - Part I
In Analysis, the real line R is usually meant as the set consisting of rational
and irrational numbers. Sometimes, it is also confused with the open interval
(0, 1), which shares many topological and measure theoretical properties with
R, since they are homeomorphic. However, in Set Theory, when one deals with
problems regarding the real line, this is usually meant in a different flavour. The
spaces which are largely investigated in the burgeoning area called set theory of
the real line are the Cantor space and the Baire space, viewed as set theoretical
privileged substitutes of R.
Definition 9. The Cantor Space 2ω is the set consisting of functions x : ω → 2,
endowed with the topology generated by basic open sets [s] = {x ∈ 2ω : s E x},
with s ∈ 2<ω , i.e., the set of finite sequences of 0s and 1s. Analogously, one
may define the Baire Space ω ω as the set of functions x : ω → ω, endowed with
the topology generated by [s] = {x ∈ ω ω : s E x}, with s ∈ ω <ω , i.e., the set of
finite sequences of natural numbers.
One can also define a notion of countably additive measure on such spaces.
This measure is induced by the measure m on basic open sets defined as follows.
Cantor space: for every s ∈ 2<ω , let m([s]) = 2−|s| , where |s| denotes
the length of s.
Q
Baire space: for every s ∈ ω <ω , let m([s]) = j<|s| 2−(s(j)+1) .
From such m, one can construct the usual outer measure with the corresponding
Carathedory’s family, so to obtain a notion of measure and the corresponding
family of measurable sets via a pretty standard method. Furthermore, once we
have a topology and a notion of measure, we can introduce the two corresponding
regularity properties.
Definition 10. Let X ⊆ 2ω .
8
• X is said to be measurable iff there is a Borel set B such that X∆B is
null;
• X is said to satisfy the Baire property iff there is a Borel set B such that
X∆B is meager.
Analogously, one can define such notions on the Baire space.
Many other notions of regularity can be introduced in a similar fashion. In
the first chapter of my PhD thesis ([4]), I use a general method to introduce
notions of regularity, of which Lebesgue measurability and Baire property turn
out to be special cases. (We acknowledge that a similar approach has been
independently used by Daisuke Ikegami and Yurii khomsky.)
We said above that 2ω and ω ω are somehow similar to the real line R. Let
us now see what we meant.
Two topological spaces X, Y are Borel isomorphic iff there is a function
f : X → Y such that for any Borel set B ⊆ Y we have f −1 [B] is Borel.
Fact 11. R, 2ω and ω ω are Borel isomorphic.
Even more, one can show that those spaces are homeomorphic up to a countable set. As an example, one may pick the function f : ω ω → 2ω \ Q (Q are
the rationals, which are those elements in 2ω which are eventually equal 0) such
that
x0
x1
xn
z }| {
z }| { z }| {
x = x0 x1 . . . xn . . . 7→ f (x) = 00 . . . 0 1 00 . . . 0 1 . . . 1 00 . . . 0 1 . . .
Borel isomorphisms are interesting from our viewpoint because questions regarding regularity properties are invariant between such spaces. More precisely,
given two topological spaces X, Y, f Borel isomorphism between them, and
X ⊆ X one has
X has the Baire property ⇔ f [X] has the Baire property.
(The same holds for Lebesgue measurability, perfect set property and any other
regularity property introduced over the years.)
For a detailed and extensive study of the real line with set theoretical
methods, we refer the reader to [2].
Forcing
Definition 12.
1. A forcing notion P is a triple (P, ≤, 1), where P is called
the set of conditions, ≤ is a partial order on P , without any minimal
element (i.e., for every p ∈ P there is q ∈ P such that q ≤ p.), and
1 represents the maximum w.r.t. ≤. Sometimes we abuse notation by
writing that a condition is in P, by identifying the forcing with its set of
conditions. Given two conditions p, q ∈ P such that q ≤ p we say that q
is stronger than p.
2. A subset D ⊆ P is said to be dense iff ∀p ∈ P ∃q ∈ D(q ≤ p).
3. A subset F ⊆ P is said to be a filter iff:
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(a) ∀p, q ∈ F ∃r ∈ F (r ≤ p ∧ r ≤ q);
(b) if p ∈ F then ∀q ≤ p(q ∈ F ).
4. A filter G is said to be generic iff for every dense subset D of P , one has
G ∩ D 6= ∅.
Further, we say that two conditions p, q ∈ P are incompatible (p ⊥ q) iff
there is no r ∈ P such that r ≤ p and r ≤ q.
The existence of a generic filter is not generally provable. However, the
following result is what we need for our purpose.
Fact 13. Let D = {Dn : n ∈ ω} be a countable family of dense subsets of P .
Then there is a filter F ⊆ P such that for every n ∈ ω, F ∩ Dn 6= ∅.
Proof. Consider the following recursive construction: pick p0 ∈ D0 , and for
every n ∈ ω, pick pn+1 ≤ pn such that pn+1 ∈ Dn+1 . Finally let F be the filter
generated by these pn ’s, i.e., F = {q ∈ P : ∃n ∈ ω(q ≤ pn )}. Then F satisfies
the required property.
Hence, the following is a simple corollary.
Corollary 14. Let M be a countable transitive model of ZFC∗ (i.e., some finite
fragment of ZFC), and let P ∈ M be a forcing notion. Then there is G ⊆ P
generic filter over M, i.e., for every dense D ⊆ P , if D ∈ M then G ∩ D 6= ∅.
The following result shows a sufficient condition for which G ∈
/ M.
Fact 15. Let M be a transitive countable model and P be a forcing notion such
that
∀p ∈ P∃q0 , q1 ∈ P(q0 ≤ p ∧ q1 ≤ p ∧ q0 ⊥ q1 ).
If G is P-generic over M, then G ∈
/ M.
The proof is by contradiction. Assume G ∈ M and put D = P \ G. Then
D ∈ M and it is easy to check that D is dense. But, G ∩ D = ∅, contradicting
G being generic.
The method of forcing was invented by Paul Cohen to build a model for
ZFC, in which CH fails. Roughly, it says what follows:
(??) Let M be a countable model for ZFCn , where the latter is a finite
conjunction φ0 ∧ · · · ∧ φn of some axioms of ZFC. Further, let P ∈ M be a
forcing notion and G ⊆ P a generic filter over M. Then we can add G to the
model M in order to define a structure M[G] as the smallest one extending M
and containing G as an element. We also obtain the following:
• M[G] |= ZFCn
• M[G] ∩ ON = M ∩ ON.
Moreover, depending on the combinatorial properties of P, one can show that
some further specific statements are true in M[G]. That means that different
filters G’s can decide differently about the truth of a given statement. But, how
this model M[G] looks like?
Very brief and rough presentation of the construction of M[G]. Our
presentation is nothing more than a very rough and informal compendium of
10
Kunen’s introduction to the method of forcing (see [3]), which represents, in our
opinion, one of the most precise and detailed expositions written so far. First
of all, we have to show how the elements of this new model are made. Given a
forcing P, people living in M will not be generally able to know what an element
in M[G] is, since that depends on the choice of the generic G. However, what
people in M may do is to have a name for elements in M[G]. Formally, a Pname can be defined recursively as follows: τ is a P-name iff τ is a relation and
and every (σ, p) ∈ τ is in τ and p ∈ P. The class of P-names in M is denoted
by MP . As we said, a name does not definitively decide an element in M[G],
but it is just a way to give a description, generally depending on the generic G,
about how it will look like in the extension M[G] (“extension” is used because
we somehow extend the model M adding the new object G). Hence, we need a
notion of evaluation of such names:
Definition 16. The evaluation of τ via G is denoted by τG and it is defined as
follows: τG = {σG : ∃p ∈ G((σ, p) ∈ τ )}. We then put
M[G] = {τG : τ ∈ MP }.
Let us now see some easy examples. The empty-set is trivially a P-name and
its evaluation is empty, by definition. Hence 0 = 0G , for any generic G. Now,
let τ = {(0, p)}, for some p ∈ P. In such a case, the evaluation of τ strictly
depends on G; in fact,
τG = 0 if p ∈
/ G, and
τG = {0} = 1 if p ∈ G.
In some very particular cases, the evaluation is independent of G; for instance
we just mentioned the case 0 = 0G . Actually, one can note that if τ = {(0, 1)},
then τG does not depend on G, since any filter contains the greatest element 1.
More generally,
if τ = {(σj , 1) : j ∈ J} then τG = {σG : j ∈ J}.
As a consequence, one obtains that there are standard names for elements of M
which are independent of G; for x ∈ M there is a name x̌ defined recursively by:
x̌ = {(y̌, 1) : y ∈ x}. By the previous argument, x̌ is independent of the generic
G ⊆ P.
There is also a canonical name for the generic G itself, i.e.,
Γ = {(p̌, p) : p ∈ P}.
In fact, by definition, ΓG = {p̌G : p ∈ G} = {p : p ∈ G} = G.
Definition 17. Given a condition p ∈ P and some sentence φ, we define as
follows:
p φ ⇔ for every generic G(p ∈ G ⇒ M[G] |= φ).
Note that, by definition, if q ≤ p and p φ, then also q φ. That explains
why we use the word stronger when q ≤ p; in fact, this is meant in the sense
that q contains more information than p, since it forces the same statements
forced by p, and possibly some more sentences.
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At first sight, it could seem that the decision whether or not some formula
φ holds in M[G] depends on the knowledge of all generic G containing p as an
element. Nevertheless, the following two facts, which are the key basic results
of the theory of forcing, say something surprising:
Fact 1. One can decide within M whether or not p φ.
Fact 2. For any formula φ, if M[G] |= φ then there is p ∈ G such that p φ.
Let us see some concrete examples of forcing notions.
Example 18. Let P(ω, ω) be the forcing consisting of finite sequence of natural
numbers, ordered by extensions. Formally,
P(ω, ω) = {p : p is a function ∧ ∃n(dom(p) = n) ∧ ran(p) ⊂ ω},
ordered by extension, i.e., q ≤ p iffSdom(q) ⊇ dom(p) and qdom(p) = p. Let G
be generic over M and put xG = {p : p ∈ G}.
Claim. xG belongs to ω ω , i.e., it is an infinite sequence of natural numbers.
Proof. We have to show two things:
1. for every n ∈ ω, n ∈ dom(p), and
2. for every n ∈ ω, n ∈ ran(p).
But, 1 and 2 simply follows by noting that
Dn = {p ∈ P(ω, ω) : n ∈ dom(p)} and Dn = {p ∈ P(ω, ω) : n ∈ ran(p)}
are dense, for every n ∈ ω. Remark that 2 is not strictly necessary to get
xG ∈ ω ω , but it serves mostly to check that xG is surjective.
A sceptic reader may now object that xG is simply a real (or better, an
element of ω ω ) belonging to the model M, and what we are doing is just a
strange way to define such a real. Let us now show that this is not the case.
Fact 19. M[G] |= ∀z ∈ ω ω ∩ M∃n ∈ ω(z(n) < xG (n)).
Before proving it, let us note that this is sufficient to show that xG ∈
/ M,
for otherwise z = xG would contradict the statement. So that means that the
forcing technique is useful to add new reals to a model of ZFC.
Proof of Fact 11. It suffices to note that, for every z ∈ ω ω ∩ M,
Dz = {p ∈ P(ω, ω) : ∃n(n ∈ dom(p) ∧ p(n) > z(n))}
is dense in P.
Example 20. Define P(ω2 × ω, 2) in a similar fashion, i.e.,
P(ω2 ×ω, 2) = {p : p is a function∧∃(α, n) ∈ ω2 ×ω((α, n) ∈ dom(p))∧ran(p) ⊆ 2}.
Note that, for every α, β ∈ ω2 , n ∈ ω,
n
Dα,β
= {p ∈ P(ω2 × ω, 2) :
∃k ≥ n((α, k) ∈ dom(p) ∧ (β, k) ∈ dom(p)
∧ p(α, k) 6= p(β, k)}
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is dense. Hence, P(ω2 × ω, 2) adds ω2 many new pairwise different elements of
2ω . This would seem to directly imply that we have proved that the extension
M[G] |= 2ω = ω2 , for any G ⊆ P(ω2 × ω, 2)-generic over M. Nevertheless,
one has to be more precise. In fact, what we have just shown is simply that
M[G] |= 2ω = (ω2 )M , where (ω2 )M represents the ω2 of M, which may be
collapsed to some smaller cardinal in M[G] (As an example of forcing notion
which certainly collapses ω2 to ω, one may think of P(ω, ω2 ), similarly defined.)
However, this is not the case of P(ω2 × ω, 2). In fact, one can show that this
forcing satisfies a property called countable chain condition, which in particular
ensures such forcing does not collapse ω2 , or in other words (ω2 )M = (ω2 )M[G] .
De facto, this forcing provides the famous result obtained by Paul Cohen in
the 1960s, to build a model for ZFC+¬CH.
Real Line - Part II
We now look at the real line with a particular emphasis on forcing method.
In the first example of forcing that we introduced in the previous section, we
have seen that P(ω, ω) adds a new real, or better, an element of ω ω . This brief
section is devoted to introduce, in a fairly general way, several notions of forcing
adding new reals. At this aim, we introduce the notion of tree. Remind that:
• 2<ω denotes the set of finite sequences (arbitrarily large) of 0s and 1s.
• ω <ω denotes the set of finite sequences (arbitrarily large) of natural numbers.
Definition 21.
• A set T ⊆ ω <ω is said to be a tree iff T is closed under
initial segments, i.e.,
∀t ∈ T ∀n ∈ ω(n < |t| ⇒ tn ∈ T ),
where tn is the restriction of t having domain equals n.
• A tree T is said to be perfect iff any of its node is extendible to a splitting
node, i.e.,
∀s ∈ T ∃t ∈ T (s E t ∧ ∃i, j(i 6= j ∧ ta i ∈ T ∧ ta j ∈ T )).
A node t as above is called splitting. Note that a perfect tree has necessarily infinite height.
• Given a tree T , one says that x ∈ ω ω is a branch through T iff ∀n ∈
ω(xn ∈ T ). The set of all branches through T is denoted by [T ] and it is
called the body of T .
(Note that all of these definitions can be similarly introduced for 2ω as well.)
The notion of tree plays an important role, since it is equivalent to the notion
of closed set. More precisely, if T is a tree, then [T ] is closed. And, if C is any
closed set, then there is a tree T such that [T ] = C (note that we did not specify
which space we are working with, since this is true both for the Baire and for
the Cantor space).
Using the notion of tree, one can now introduce a rather general notion of
forcing.
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Definition 22. A forcing P is said to be arboreal iff every T ∈ P is a perfect
tree and ∀t ∈ T (Tt ∈ P), where Tt := {s ∈ T : s E t ∨ t E s}.
Example 23.
1. Forcing P(ω, ω) can be viewed as an arboreal forcing. In
fact, given a condition p ∈ P(ω, ω), one can define Tp = {t ∈ ω <ω : p E t}.
Then let C = {Tp : p ∈ P(ω, ω)}, ordered by inclusion, i.e., T 0 ≤ T iff
T 0 ⊆ T . It is straightforward to check that C and P(ω, ω) are isomorphic,
and hence they yield the same forcing extension. (This fact can be rather
easily proved.) Letter C to denote such forcing comes from the fact that
this is nothing more than the forcing introduced by Cohen in the 1960s.
It turns out that C is strictly connected with the Baire property.
2. Consider the forcing R = {T ⊆ 2<ω : [T ] has strictly positive measure},
ordered by inclusion. This forcing is usually called random forcing, and its
raison d’être consists of the strict connection with Lebesgue measurability.
3. Sacks forcing S = {T ⊆ 2ω : T is a perfect tree}, ordered by inclusion.
4. Silver forcing V = {T ∈ S : ∀s, t ∈ T (|t| = |s| ⇒ (ta i ∈ T ⇔ sa i ∈ T ))},
ordered by inclusion.
5. Miller forcing M = {T ⊆ ω ω : T is a perfect tree whose splitnodes have
infinitely many successors}, ordered by inclusion.
One can also introduce a notion of smallness for subsets of the real line
associated with a certain arboreal forcing P, and thus its corresponding notion
of regularity.
Definition 24.
1. A set of reals X is said to be P-null iff
∀T ∈ P∃T 0 ∈ P(T 0 ⊆ T ∧ [T 0 ] ∩ X = ∅).
Then, let JP = {X : X is P-null}, and
S IP be the closure of JP under
countable unions, i.e., IP = {X : X = n∈ω Xn , for Xn ’s in JP }.
2. A set of reals X is said to be P-measurable iff
∀T ∈ P∃T 0 ∈ P(T 0 ⊆ T ∧ ([T 0 ] ∩ X ∈ IP ∧ [T 0 ] \ X ∈ IP )).
One can prove (rather surprisingly?) that C-measurability coincides with the
Baire property, while R-measurability coincides with Lebesgue measurability.
Historical Background and recent results
The interest of Mathematics around regularity properties is rather old. The precursors may be considered the Lebesgue measurability and the Baire property.
Assuming AC (Axiom of Choice), Vitali was able to construct a non-measurable
set, and the same proof also provided a set without Baire property. So a natural question was to understand whether or not AC was really necessary to
obtain such a nasty set. In the 1960s, Solovay resolved this problem, by using
the method of forcing. Solovay’s forcing may be viewed as a generalization of
Cohen forcing previously introduced. Given two cardinals γ < λ, one defines
P(γ, λ) = {p : p is a function ∧ ∃α < γ(dom(p) = α ∧ ran(p) ⊂ λ},
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ordered by extension. One can show that P(γ, λ) collapses λ to γ. More generally, given κ > γ, one can define the product forcing
Y
P(γ, < κ) =
P(γ, λ).
λ<κ
In the generic extension via this product forcing, one can show that any cardinal
strictly below κ is collapsed to γ, and so κ becomes γ + .
Hence, denoting by V[G] the extension via P(ω, < κ), with κ inaccessible,
Solovay established the following result:
V[G] |= “all sets of reals in L(R) are Lebesgue measurable”
and so in particular L(R)V [G] |= “all sets of reals are Lebesgue measurable”.
The same also held for the Baire property. Solovay’s model, introduced in [9],
represented (and still represents nowadays) one of the corner-stones of set theory
of the reals. Furthermore, Solovay’s model turned out to be the tip of the iceberg
of a rich research field.
In the 1980s, Shelah resolved the most intriguing problem risen from Solovay’s work. In [7], Shelah introduced a really profound construction, called
amalgamation, to build strongly homogeneous algebras, by mean of which he
was able to establish the following result: starting from a ground model V without inaccessible cardinals, one can build a Boolean algebra B, such that V[G] |=
“all sets of reals in L(R) have the Baire property”, where G is B-generic over
V. On the contrary, yet in the same paper, Shelah also established: if we assume “all Σ13 sets are Lebesgue measurable”, then one can prove that for every
x ∈ ω ω , L[x] |= “ω1 is inaccessible”.
Hence, Shelah’s results marked a huge difference between the behaviour of
the Baire property and that one of the Lebesgue measurability, by showing
that, whereas the latter does need an inaccessible, the former does not. Note
also that, as a consequence, one obtains that in Shelah’s model where all sets of
reals have the Baire property, there exists a non-measurable set. Such a result
was the first one regarding separation of regularity properties on the family of
all sets of reals; furthermore, just one year later (in [8]) Shelah also showed the
reverse separation, by constructing a model where all sets of reals are Lebesgue
measurable, but there exists a set without Baire property.
In the wake of these results, in [4], Friedman and I constructed a model
where
• all sets are Silver-measurable;
• there is a set which is not Miller-measurable;
• ω1 is inaccessible by reals.
This results has been then improved in [5], by introducing the notion of unreachability, where I obtained a model for
• all sets are Silver-measurable;
• there is a non-Miller-measurable set;
• there is a non-Lebesgue-measurable set;
• ω1 is inaccessible by reals.
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References
[1] Alessandro Andretta, Dispense corso “Istituzioni di Logica Matematica”.
[2] Tomek Bartoszyński, Haim Judah, Set Theory-On the structure of the real
line, AK Peters Wellesley (1999).
[3] Kenneth Kunen, Set Theory-An introduction to independence proof, Elsevier,
1980.
[4] Giorgio Laguzzi, Arboreal forcing notions and regularity properties of the
real line, PhD thesis, Universität Wien, 2012, written under the supervision
of Sy D. Friedman.
[5] Giorgio Laguzzi, On the separation of the regularity properties of the real
line, submitted.
[6] Gabriele Lolli, Dagli insiemi ai numeri, Bollati Boringhieri, 1995
[7] Saharon Shelah, Can you take Solovay’s inaccessible away?, Israel Journal
of Mathematics, Vol. 48 (1985), pp 1-47.
[8] Saharon Shelah, On measure and category, Israel Journal of Mathematics,
Vol. 52 (1985), pp 110-114.
[9] Robert M. Solovay, A model of set theory in which every set of reals is
Lebesgue measurable, Annals of Mathematics, Vol. 92 (1970), pp 1-56.
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