Axiomatic Set Theory
Alexandru Baltag
(ILLC, University of Amsterdam)
LECTURE NOTES 11: The Constructibe Universe
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Relativization of a formula to a set
Let M be a transitive set (i.e. such that x ∈ y ∈ M implies x ∈ M ).
For every formula ϕ of the language LAST, such that all the names
(constants) w1 , . . . , wn occuring in ϕ denote sets a1 , . . . , an ∈ M , we
define the relativization ϕM of ϕ to the set M by recursion:
(vi ∈ vj )M = (vi ∈ vj ),
(¬ϕ)M = (¬ϕM ),
(vi = vj )M = (vi = vj )
(ϕ∧ψ)M = (ϕM ∧ψ M ),
(∀xϕ)M = ∀x(x ∈ M ⇒ ϕM ),
(ϕ∨ψ)M = (ϕM ∨ψ M )
(∃xϕ)M = ∃x(x ∈ M ∧ ϕM )
Essentially, ϕM is obtained from ϕ simply by restricting the quantifiers
to M .
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Transitive Models of LAST
In this way, a transitive set M can be taken to be a model of the
language of Set Theory:
We say that a formula ϕ of LAST is true, or satisfied, in a
transitive model M , and we write M |= ϕ, iff the formula ϕM is true.
A transitive set M is a model of the system ZF (or ZF C) iff all
the axioms of ZF (or ZF C) are satisfied in it,
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Class Models
The same definitions can be extended to transitive classes M :
in this way, we can talk about satisfaction of a formula in a transitive
class, and introduce transitive class-models of ZF C etc.
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Interpretation of a set-theoretic class in a model
Given a transitive model M (set or class), and given a definable class
C = {x|ϕ(x)}
defined by some formula ϕ(v), having only one free variable v and such
that all names w1 , . . . , wn occuring in ϕ denote sets a1 , . . . , an ∈ M , we
can define the interpretation C M of class C in model M :
C M = {x ∈ M |ϕM (x)}.
Example: The interpretation of the “universe” class
V = {x|x = x}
is always the collection of all the sets in the model:
V M = M.
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Definability over a transitive set/class
Given a transitive set (or class) M , we say that a subset M 0 ⊆ M is
definable over M iff M 0 is the interpretation in M of some class
(definable by some formula of LAST);
more precisely, iff there exists some formula
ϕ(v)
of the language LAST, having only one free variable v, such that all the
names (constants) w1 , . . . , wn occuring in the formula ϕ(v) denote sets
a1 , . . . , an ∈ M , and such that we have
M 0 = {x|ϕ(x)}M ,
i.e.
M 0 = {x ∈ M |ϕM (x)}.
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The collection of definable subsets
The set (or class)
Def (M ) = {x ⊆ M |x definable over M }
of all the sets definable over M is typically a proper subset
Def (M ) ⊂ P(M )
of the powerset of M .
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The Constructible Hierarchy
We can now replace the classical iterative hierarchy of “universes”
hVα |α ∈ Oni
by a more restricted hierarchy
hLα |α ∈ Oni,
called the constructible hierarchy.
By recursion, at each successor step we collect only ALL DEFINABLE
subsets of the previous “universe” (instead of ALL subsets):
L0 = ∅ ,
Lα+1 = Def (Lα ) ,
[
Lλ =
Lα , for λ limit .
α<λ
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Properties of the Constructible hierarchy
The constructible hierarchy share some properties in common with the
classical iterative hierarchy:
Lα ⊆ Lβ for α ≤ β ,
every Lα is transitive ,
Lα ∩ On = α .
BUT there are major differences, e.g.
|Lα | = |α| for every infinite α
(since the language LAST is countable), and hence
|Lω+1 | = ℵ0 < |Vω+1 | = 2ℵ0 .
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The Constructible Universe and Axiom of Constructibility
Put
L :=
[
Lα
α
This (proper) class is called the constructible universe.
QUESTION: Are there are any sets that are NOT constructible?
The so-called “Axiom of Constructibility”
V =L
asserts that the answer is NO.
This axiom is a new principle, independent of ZF C. But we’ll show
that it is consistent with ZF (C) and that it implies the Axiom of
Choice (AC)! This proves that AC is consistent with ZF .
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Does this make any sense?
But how could we possibly have V = L??
For instance, we would then have that P(ω) ∈ L.
This worked in V , since we had
ω ⊆ Vω
hence
P(ω) ⊆ P(Vω ) = Vω+1
and thus
P(ω) ∈ P(Vω+1 ) = Vω+2 .
BUT in contrast
P(ω) 6⊆ Lω+1 (since |Lω+1 | = ℵ0 )!
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NEW Subsets of ω Keep Appearing
The point is that, unlike in the classical hierarchy, in the constructible
hierarchy new subsets of ω keep being created at higher and higher
stages.
Only the “simplest” subsets of ω are in Lω+1 .
More complex ones (having a more complex defining formula) appear
only in Lω+2 , etc.
But, as we get to higher and higher ordinals, it is POSSIBLE that “all”
subsets of ω will EVENTUALLY occur.
Indeed, the Axiom of Constructibility implies that this actually will
happen.
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Axiom of Constructibility is Consistent with ZF
Is V = L consistent with the other axioms of Set Theory?
THEOREM 5.4.1 (Devlin) If ZF (without AC) is consistent then
ZF + (V = L) is consistent.
PROOF: Let M be a model of ZF (which, for simplicity, we assume to
be transitive). Simply repeat the construction of L INSIDE M : i.e.
construct the interpretation LM of the constructible universe L in the
model M . Using the (fact that M satisfies the) axioms of ZF , we can
show that LM also satisfies them.
Moreover, we can check that in the model LM all sets are constructible.
So the interpretation of L inside model LM is the whole model:
(LM )
L
=L
M
=V
(LM )
Hence, the axiom L = V is satisfied in the model LM .
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Relative Consistency of AC
THEOREM 5.3.1 (Devlin):
ZF + (V = L) ⇒ AC .
PROOF: Construct, by recursion on α, a family
(≤α )α∈On
of relations, such that
≤α is a well-ordering of Lα ,
and such that, for all β < α, we have
≤α |Lβ =≤β , and Lβ is an initial segment of Lα (w.r.t. the order ≤α ).
Corollary: If ZF (without AC) is consistent then ZF C is
consistent.
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Relative Consistency of GCH
THEOREM 5.5.1 (Devlin):
ZF + (V = L) ⇒ GCH .
PROOF (sketch): We only show CH. It is easy to see that
ω ⊆ Lω .
The problem is (as already noticed) that NOT ALL SUBSETS of ω are
in Lω+1 : new subsets of ω keep appearing at further stages, through
Lω+1 , Lω+2 , . . . etc.
BUT one can show that they ALL appear BEFORE the stage ω1 . So:
P(ω) ⊆ Lω1
and hence
2ℵ0 = |P(ω)| ≤ |Lω | = |ω1 | = ℵ1 .
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