Download Axiomatic Set Theory Alexandru Baltag (ILLC, University of

Axiomatic Set Theory
Alexandru Baltag
(ILLC, University of Amsterdam)
LECTURE NOTES 10: Large Cardinals and Other Topics
in Pure Set Theory
1
Inaccessible Cardinals
A cardinal κ is weakly inaccessible iff it is an uncountable, regular
limit cardinal.
A cardinal is (strongly) inaccessible iff it is an uncountable regular
cardinal satisfying
2λ < κ for all λ < κ .
Every inaccessible cardinal is also weakly inaccessible.
Moreover, if we assume GCH, then the two notions coincide:
GCH
⇒
“ every weakly inaccessible cardinal is inaccessible”.
But the contrary is also consistent with ZF C:
in fact, it is consistent with ZF C that 2ℵ0 is weakly inacessible
(although it is obviously NOT inaccessible!).
2
Inaccessible Cardinals as Models for Set Theory
The existence of inaccessible cardinals implies the consistency
of ZF C:
indeed, “κ is an inaccessible cardinal” is equivalent to “Vκ is a
model of ZF C ”.
It follows from this that, if ZF C is consistent, then the existence
of inaccessible cardinals is independent of ZF C.
3
Large Cardinal Assumptions
Since most people believe that ZF C is consistent, they conclude that it
is consistent to assume the existence of inaccessible cardinals.
The assertion of this existence is an example of “strong infinity
axiom”, or “LARGE CARDINAL” hypothesis.
There are many other examples.
How we can we justify them?
4
Reflection Principles
The observation that Vκ is a model of ZF C, for κ inaccessible, can be
made into a justification:
The Reflection Principle(s):
For every unary property P (x) that we can define (in a given
language), if we can show that P (V ) holds, then there must
exist some cardinal κ such that P (Vκ ) holds.
Essentially, this says that “the universe of all sets is undefinable”:
it cannot be uniquely characterized by any sentence in the
language.
This resembles Cantor’s conception of the “absolute infinity” of
paradoxical sets.
5
Restriction
The specific implementation depends on the language (and on P ).
Of course, the property P (x) cannot just be any property: its
quantifiers have to be all restricted to x, e.g. ∀y ∈ x.
Otherwise, we could take
P (x) = ∀y(y ∈ x)
and, since V has this property, we would conclude that there exists a
cardinal κ such that Vκ is the set of all sets!
6
Other Properties of Inaccessible Cardinals
Lemma 3.10.1. (Devlin)
If ℵκ is a weakly inaccessible cardinal, then (κ is a cardinal and) ℵκ = κ.
Theorem 3.10.2. (Devlin)
If κ is an inaccessible cardinal, then
X
κλ = κ .
λ<κ
Theorem 3.10.3. (Devlin)
Asumme GCH. A limit cardinal κ is inaccessible if and only if
X
κλ = κ .
λ<κ
7
Topologies
A topology on a set X is given by a family T ⊆ P(X) of subsets of X,
such that
• ∅ ∈ T and X ∈ T ;
• any union of (possibly infinitely many) sets in T : i.e.
S
if A ⊆ T is a set then A ∈ T ;
• any intersection of two (or finitely many) sets in T is in T : i.e.
if A, B ∈ T then A ∩ B ∈ T .
The sets A ∈ T are called open.
Their complements X − A (with A ∈ T ) are called closed.
8
Examples
The standard topology of real numbers:
X = R, T = { countable unions of open intervals }:
i.e. A ⊆ R is open iff it is of the form
A=
∞
[
(ai , bi ) , with ai , bi ∈ R.
i=1
The standard topology on Rn :
X = Rn , T = { countable unions of open “balls” }:
i.e. A ⊆ Rn is open iff it is of the form
A=
∞
[
{x ∈ Rn |d(x, ai ) ≤ bi } ,
i=1
where ai , bi ∈ Rn and d is the Euclidean distance in n-dimensional
space Rn .
9
Other Examples: the Baire Space
The Baire Space has
X = N ω = {infinite sequences ~n = (n1 , n2 , . . .) of natural numbers},
and T = { countable unions of “fans” },
i.e. A ⊆ Rn is open iff it is of the form
A=
∞
[
[~
ei ],
i=1
where each e~i is a finite sequence e~i = (e1i , e2i , . . . eji ) of natural
numbers, and
[~
ei ] := {~n ∈ X|nk = eki for all 1 ≤ k ≤ j}
is called the fan with handle e~i .
10
Cantor Space
Cantor space is similar, except that
X = 2ω = {infinite sequences ~n = (n1 , n2 , . . .) of “bits” ni ∈ {0, 1}},
with the same topology as the Baire space.
11
Boolean Algebras
A Boolean algebra is a set B together with constants 0, 1 ∈ B, a unary
operation − : B → B (“complementation”) and two binary operations
∧ : B × B → B (“meet”) and ∧ : B × B → B (“join”), satisfying
b ∧ c = c ∧ b,
b ∨ c = c ∨ b,
b ∧ (c ∧ d) = (b ∧ c) ∧ d,
b ∨ (c ∨ d) = (b ∨ c) ∨ d,
(b ∧ c) ∨ c = c,
(b ∨ c) ∧ c = c,
b ∧ (c ∨ d) = (b ∧ c) ∨ (b ∧ d),
b ∨ (c ∧ d) = (b ∨ c) ∧ (b ∨ d),
b ∧ 1 = b,
b ∨ 0 = b,
b ∧ −b = 0,
b ∨ −b = 1.
Definition: In a Boolean algebra, we put b ≤ c
12
⇐⇒
b ∨ c = c.
Examples
B = {0, 1}, with the operations being the standard logical operations
with truth values.
A (Boolean) algebra of sets (or a field of sets) is a Boolean algebra
B ⊆ P(X) consisting of subsets of a given set X, with
0 := ∅,
1 := X,
−b = X − b (set-complement),
b ∧ c = b ∪ c (union),
b ∨ c = b ∩ c (intersection).
In an algebra of sets, the order is given by inclusion:
b ≤ c iff b ⊆ c.
13
Sup (infinite join) and inf (infinite meet)
An element b ∈ B of a Boolean algebra B is the supremum (“join”) of
W
a set A ⊆ B of elements, written b = A, if b is the “least” element
that is greater or equal to all the elements of A; i.e. we have
∀a ∈ A(a ∈ A ⇒ a ≤ b), and
0
0
0
∀b ∈ B ( ∀a ∈ A(a ∈ A ⇒ a ≤ b ) ⇒ b ≤ b ) .
Similarly, b ∈ B of a Boolean algebra B is the infimum (“meet”) of a
V
set A ⊆ B of elements, written b = A, if b is the “greatest” element
that is smaller or equal to all the elements of A; i.e. if we have
∀a ∈ A(a ∈ A ⇒ a ≥ b), and
∀b0 ∈ B ( ∀a ∈ A(a ∈ A ⇒ a ≥ b0 ) ⇒ b ≥ b0 ) .
14
IF THEY EXIST, infimum and supremum are UNIQUE.
Moreover, thet conincide the Boolean join and meet in the case that the
set A is finite.
15
σ-algebras and Complete Algebras
A σ-algebra is a Boolean algebra B in which every COUNTABLE set
A ⊆ B of elements has a supremum (and also an infimum).
A complete algebra is a Boolean algebra B in which EVERY set
A ⊆ B of elements has a supremum (and also an infimum).
A measurable space (or σ-field) over a given set X is a σ-algebra
of subsets of X, in which (complementation is set-complementation
and) supremum is given by set union:
[
supA =
A.
16
Borel Sets
Given a topological space (X, T ), the Borel algebra on X is the
smallest σ-field of sets that includes all the open sets;
i.e. the smallest family B ⊆ P(X) such that T ⊆ B and such that B is
closed with respect to set-complementation and to countable unions.
The elements of B are called Borel sets.
17
Borel Hierarchy
They form a hierarchy (the “Borel hierarchy”), in the following sense:
define by recursion
B0 := {Y ⊆ X|Y ∈ T or (X − Y ) ∈ T }
(the family consisting of all open and all closed sets),
Bα+1 := { countable unions and countable intersections of members of Bα }
and
Bλ :=
[
Bα ,
for λ limit ordinal .
α<λ
Then we have
B=
[
α<ω1
18
Bα .
Measures on Boolean Algebras
A real measure on a Boolean algebra B is a function µ : B → [0, 1],
such that
µ(0) = 0, µ(1) = 1, and
if b ∧ c = 0 then µ(b ∨ c) = µ(b) + µ(c).
A 2-valued measure is similar, except that µ : B → {0, 1}.
A (ω-additive) measure on a σ-algebra B is a measure
µ : B → [0, 1], such that, for every countable set A = {an |n ∈ N } ⊆ B of
“mutually disjoint” elements (i.e. such that n 6= m implies
an ∧ am = 0), we have that
_
X
µ( an ) =
µ(an ).
n
n
A κ-additive measure is one satisfying the same additivity condition
19
for all sets A of size < κ.
20
Examples: measures on σ-fields of sets
The standard example is the Lebesgue measure on R.
21
Measure Algebras
A measure algebra is a Boolean algebra B with a with a countably
additive positive measure;
i.e. with a real-valued function on B such that:
m(0) = 0, m(1) = 1,
m(x) > 0 for x 6= 0,
m is ω-additive.
22
Probability Measures give rise to Measure Algebras
A probability measure on a measure space gives a measure
algebra on the Boolean algebra of measurable sets modulo
sets of measure 0.
For example, given a topological space (X, T ), take Borel algebra
B ⊆ P(X) on X, and take an ω-additive measure µ on B.
Now identify Borel sets that differ by a null set: i.e. take the “ideal”
∆ = {B ∈ B|µ(B) = 0},
and form the “quotient”
B = B/∆,
consisting of equivalence classes with respect to the equivalence
B ∼ B 0 iff µ((B − B 0 ) ∪ (B 0 − B)) = 0.
Then B is a measure algebra!
Countable Chain Property
Every measure algebra B has the “countable chain property”:
i.e.
there is no uncountable family F ⊆ B of pairwise “disjoint”
element of B.
Here, “pairwise disjoint” means that
∀b, b0 ∈ B : b ∧ b0 = 0.
24
Trees
A tree is a poset (T, <T ) such that, for every x ∈ T , the set
x̂ := {y ∈ T |y <T x}
of all predecessors of x is well-ordered by <T .
The height of an element x ∈ T is the ordinal number
ht(x) := Ord(x̂, <T )
For α ∈ On, the αth level of the tree T is the set
Tα = {x ∈ T |ht(x) = α}.
The height of the tree T is the least ordinal λ such that Tλ = ∅.
A branch in T is a subset of T that is linearly ordered by <T and
closed under predecessors.
25
The Tree Property
A cardinal κ has the tree property if: a tree of height κ, whose every
level is of size < κ, has a branch of length κ.
26
Konig’s Tree Lemma
Lemma. ℵ0 = ω has the Tree Property”; i.e.: a tree of height ω, whose
every level is finite, has an infinite branch.
27
Weakly Compact Cardinals
A cardinal κ is weakly compact if it is uncountable and has the tree
property.
28
Equivalent Definitions
For λ ≥ κ, a language L is called (λ, κ)-compact if: every set of
sentences of size at most λ is consistent whenever all its subsets of size
< κ are consistent.
This generalizes the Compactness Theorem of first order logic.
PROPOSITION. The following are equivalent:
1. κ weakly compact;
2. Lκ,ω is (κ, κ)-compact;
3. Lκ,κ is (κ, κ)-compact.
Here, Lκ,ω is the extension of first-order logic that allows for infinite
V
conjunctions Φ of sets of sentences Φ of size |Φ| < κ.
Lκ,κ is just like Lκ,ω , except that it also allows for quantifies ∀X over
sets X of variables of size |X| < κ.
29
Measurable Cardinals
A cardinal κ is measurable if there exists a κ-additive 2-valued
measure defined on the full powerset algebra P(κ).
30
Strongly Compact Cardinals
κ is strongly compact if Lκ,ω is (λ, κ)-compact for EVERY λ ≥ κ.
strongly compact ⇒ measurable ⇒ weakly compact ⇒ inaccessible .
All the implications are STRICT: the converses are false.
IN FACT:
κ weakly compact ⇒ there exist κ inaccessibles < κ.
31