Some compactness principles in set theory Radek Honzik Department - - PowerPoint PPT Presentation

Some compactness principles in set theory

Radek Honzik

Department of Logic Charles University logika.ff.cuni.cz/radek

PhDs in Logic May 2, 2018

• R. Honzik

Some compactness principles

Contents

We will discuss two examples of compactness principles in set theory which are motivated, or have a connection, to logical notions:

• R. Honzik

Some compactness principles

Contents

We will discuss two examples of compactness principles in set theory which are motivated, or have a connection, to logical notions: Compactness of infinitary propositional logics, or equivalently the existence of cofinal branches in certain trees.

• R. Honzik

Some compactness principles

Contents

We will discuss two examples of compactness principles in set theory which are motivated, or have a connection, to logical notions: Compactness of infinitary propositional logics, or equivalently the existence of cofinal branches in certain trees. Reflection of L¨

• wenheim-Skolem (LS) arguments to

subdomains of smaller sizes.

• R. Honzik

Some compactness principles

Contents

We will discuss two examples of compactness principles in set theory which are motivated, or have a connection, to logical notions: Compactness of infinitary propositional logics, or equivalently the existence of cofinal branches in certain trees. Reflection of L¨

• wenheim-Skolem (LS) arguments to

subdomains of smaller sizes. In general, a certain system S is compact with respect to a property P if from the fact that all “small” parts of the system S have P, we can conclude that S has P.

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Some compactness principles

Trees: definitions

We say that a partial order (T, <) is a tree if the set of <-predecessors of every t ∈ T is wellordered by <. Thus tree is a generalization of a wellordered set (wellorder implies linearity). If t is in T, we denote by ht(t)T the height of the node t in T: the

• rdinal-type of the wellordering of the predecessors of t.
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Some compactness principles

Trees: definitions

We say that a partial order (T, <) is a tree if the set of <-predecessors of every t ∈ T is wellordered by <. Thus tree is a generalization of a wellordered set (wellorder implies linearity). If t is in T, we denote by ht(t)T the height of the node t in T: the

• rdinal-type of the wellordering of the predecessors of t.

For every α, let LevT(α) be the set of all t ∈ T with ht(t)T = α.

• R. Honzik

Some compactness principles

Trees: definitions

We say that a partial order (T, <) is a tree if the set of <-predecessors of every t ∈ T is wellordered by <. Thus tree is a generalization of a wellordered set (wellorder implies linearity). If t is in T, we denote by ht(t)T the height of the node t in T: the

• rdinal-type of the wellordering of the predecessors of t.

For every α, let LevT(α) be the set of all t ∈ T with ht(t)T = α. The height of tree T, denoted ht(T), is the least α such that LevT(α) is empty.

• R. Honzik

Some compactness principles

Trees: definitions

We say that a partial order (T, <) is a tree if the set of <-predecessors of every t ∈ T is wellordered by <. Thus tree is a generalization of a wellordered set (wellorder implies linearity). If t is in T, we denote by ht(t)T the height of the node t in T: the

• rdinal-type of the wellordering of the predecessors of t.

For every α, let LevT(α) be the set of all t ∈ T with ht(t)T = α. The height of tree T, denoted ht(T), is the least α such that LevT(α) is empty. T is called a κ-tree, where κ is a cardinal, if ht(T) = κ, and for all α < ht(T), |LevT(α)| < κ.

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Some compactness principles

Trees: examples

Every ordinal α is a tree with the ordering given by ∈, and its height is α. If α is a cardinal, then α is an α-tree.

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Some compactness principles

Trees: examples

Every ordinal α is a tree with the ordering given by ∈, and its height is α. If α is a cardinal, then α is an α-tree. For every ordinal α, let <α2 be the set of all functions t : β → 2, for β < α. The pair T = (<α2, ⊆) is a tree of height α, which we call the full binary tree. If α is a cardinal, then T is an α-tree iff α is a strong limit cardinal (i.e. 2µ < α for every µ < α).

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Some compactness principles

Trees: examples

Every ordinal α is a tree with the ordering given by ∈, and its height is α. If α is a cardinal, then α is an α-tree. For every ordinal α, let <α2 be the set of all functions t : β → 2, for β < α. The pair T = (<α2, ⊆) is a tree of height α, which we call the full binary tree. If α is a cardinal, then T is an α-tree iff α is a strong limit cardinal (i.e. 2µ < α for every µ < α). In general, (<αβ, ⊆) is a tree of height α in which each nodes splits into |β|-many successors.

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Some compactness principles

Trees: examples

Every ordinal α is a tree with the ordering given by ∈, and its height is α. If α is a cardinal, then α is an α-tree. For every ordinal α, let <α2 be the set of all functions t : β → 2, for β < α. The pair T = (<α2, ⊆) is a tree of height α, which we call the full binary tree. If α is a cardinal, then T is an α-tree iff α is a strong limit cardinal (i.e. 2µ < α for every µ < α). In general, (<αβ, ⊆) is a tree of height α in which each nodes splits into |β|-many successors. We say that T ⊆ <αβ together with the inclusion relation is a tree if it is closed under initial segments.

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Some compactness principles

Trees: branches

Let (T, <) be a tree. We say that B ⊆ T is a branch if it is linearly ordered by < and is maximal with respect to inclusion. We say that a branch B is cofinal in T if B ∩ LevT(α) is non-empty for every α < ht(T).

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Some compactness principles

Trees: branches

Let (T, <) be a tree. We say that B ⊆ T is a branch if it is linearly ordered by < and is maximal with respect to inclusion. We say that a branch B is cofinal in T if B ∩ LevT(α) is non-empty for every α < ht(T). The tree T = (<ω2, ⊆) has 2ω-many cofinal branches (and all its branches are cofinal). With the usual identification of real numbers with subsets of ω, it follows that the real numbers R can be identified with the branches in T (with this identification we call ω2 the Cantor space). Notice that T is an ω-tree since all its levels are finite.

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Some compactness principles

Trees: branches

Let (T, <) be a tree. We say that B ⊆ T is a branch if it is linearly ordered by < and is maximal with respect to inclusion. We say that a branch B is cofinal in T if B ∩ LevT(α) is non-empty for every α < ht(T). The tree T = (<ω2, ⊆) has 2ω-many cofinal branches (and all its branches are cofinal). With the usual identification of real numbers with subsets of ω, it follows that the real numbers R can be identified with the branches in T (with this identification we call ω2 the Cantor space). Notice that T is an ω-tree since all its levels are finite. The tree T = (<ωω, ⊆) is not an ω-tree and its (all cofinal) branches are elements of ωω, which we call the Baire space.

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Some compactness principles

Trees: branches

For every k < ω, let Ak denote the set of all pairs (m, k) such that m < k and consider the set T =

k<ω Ak. Order T by

defining (m, k) < (m′, k′) iff m < m′ and k = k′. Then (T, <) is a tree of height ω which does not have a cofinal branch. Note that T has levels of size ω, so T is not an ω-tree.

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Some compactness principles

Trees: branches

For every k < ω, let Ak denote the set of all pairs (m, k) such that m < k and consider the set T =

k<ω Ak. Order T by

defining (m, k) < (m′, k′) iff m < m′ and k = k′. Then (T, <) is a tree of height ω which does not have a cofinal branch. Note that T has levels of size ω, so T is not an ω-tree. Let T the set of all strictly increasing sequences of rational numbers with the greatest element. Then (T, ⊆) is a tree of height ω1 which has no cofinal branch. Note that T has levels

• f size 2ω, and so T is not an ω1-tree.
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Some compactness principles

Question

Question: If κ ≥ ω is a regular cardinal, does every κ-tree have a cofinal branch?

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Some compactness principles

Question

Question: If κ ≥ ω is a regular cardinal, does every κ-tree have a cofinal branch? Note that if κ is singular, then it is easy to build a tree T of height κ whose levels have size at most cf (κ) < κ which does not have a cofinal branch. Thus the limitation to regular cardinals is without the loss of generality (in other words, the question has the trivial answer no for singular cardinals κ).

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Some compactness principles

ω-trees

Let us first deal with κ = ω.

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Some compactness principles

ω-trees

Let us first deal with κ = ω. Theorem (K¨

• nig)

Every ω-tree has a cofinal branch.

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Some compactness principles

ω-trees

Let us first deal with κ = ω. Theorem (K¨

• nig)

Every ω-tree has a cofinal branch. Proof. The proof is by induction, and crucially uses the fact that there are no limit stages in the inductive proof.

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Some compactness principles

Compactness of propositional logic

K¨

• nig’s lemma implies the compactness for the classical

propositional logic, which we denote PL(ω).

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Some compactness principles

Compactness of propositional logic

K¨

• nig’s lemma implies the compactness for the classical

propositional logic, which we denote PL(ω). Details: Suppose E is a family of infinitely many propositional

• formulas. Let E0 ⊆ E1 ⊆ · · · be finite sets whose union is E and

let e0 < e1 < · · · be strictly increasing natural numbers such that all variables in En are among the first en variables. Let Tn be the collection of all s ∈ en2 which satisfy all formulas in En. Let T be the union of the Tn’s closed under initial segments (T is an ω-tree and a subtree of <ω2). By construction, any cofinal branch in T gives an evaluation of all variables and this evaluation satisfies all formulas in E.

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Some compactness principles

Infinitary logics and trees

Let κ ≥ ω be a regular cardinal and let us consider a propositional logic PL(κ) which has κ-many variables and allows formulas of size < κ (composed of the usual propositional connectives).1 Notice that PL(ω) is the usual propositional logic.

1We can consider infinitary predicate logic Lκ,κ but we will not get more

strength.

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Some compactness principles

Infinitary logics and trees

Let κ ≥ ω be a regular cardinal and let us consider a propositional logic PL(κ) which has κ-many variables and allows formulas of size < κ (composed of the usual propositional connectives).1 Notice that PL(ω) is the usual propositional logic. We say that a set of propositional formulas F in PL(κ) is κ-satisfiable if every E ⊆ F of size < κ is satisfiable. PL(κ) is compact if every κ-satisfiable F is satisfiable.

1We can consider infinitary predicate logic Lκ,κ but we will not get more

strength.

• R. Honzik

Some compactness principles

Infinitary logics and trees

Let κ ≥ ω be a regular cardinal and let us consider a propositional logic PL(κ) which has κ-many variables and allows formulas of size < κ (composed of the usual propositional connectives).1 Notice that PL(ω) is the usual propositional logic. We say that a set of propositional formulas F in PL(κ) is κ-satisfiable if every E ⊆ F of size < κ is satisfiable. PL(κ) is compact if every κ-satisfiable F is satisfiable. As we discussed above, <κ2 is a κ-tree iff κ is strong limit. With the regularity of κ, which we assume, <κ2 is a κ-tree iff κ is an inaccessible cardinal. It follows that when κ is inaccessible, the set

• f all evaluations of < κ-many variables in PL(κ) is a κ-tree.

1We can consider infinitary predicate logic Lκ,κ but we will not get more

strength.

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Some compactness principles

Compactness of an inaccessible κ

The following theorem captures the connection between κ-trees and PL(κ). Instead of saying that every κ-tree has a cofinal branch, we often say that κ has the tree property, TP(κ).

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Some compactness principles

Compactness of an inaccessible κ

The following theorem captures the connection between κ-trees and PL(κ). Instead of saying that every κ-tree has a cofinal branch, we often say that κ has the tree property, TP(κ). Theorem Suppose that κ is an inaccessible cardinal. Then the following statements are equivalent: PL(κ) is compact. Every κ-tree has a cofinal branch, TP(κ).

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Some compactness principles

Compactness of an inaccessible κ

The following theorem captures the connection between κ-trees and PL(κ). Instead of saying that every κ-tree has a cofinal branch, we often say that κ has the tree property, TP(κ). Theorem Suppose that κ is an inaccessible cardinal. Then the following statements are equivalent: PL(κ) is compact. Every κ-tree has a cofinal branch, TP(κ). A cardinal κ with PL(κ) being compact is called weakly compact (a relatively mild large cardinal concept, but far stronger than inaccessibility).

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Some compactness principles

Compactness for successor cardinals

If κ is not inaccessible (but stays regular), the set of all evaluations

• f < κ-many variables is no longer a κ-tree. And we know that

there are always trees of height κ which are not κ-trees and have no cofinal branches. A meaningful notion of compactness for a successor κ must therefore be formulated more carefully.

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Some compactness principles

Compactness for successor cardinals

If κ is not inaccessible (but stays regular), the set of all evaluations

• f < κ-many variables is no longer a κ-tree. And we know that

there are always trees of height κ which are not κ-trees and have no cofinal branches. A meaningful notion of compactness for a successor κ must therefore be formulated more carefully. Definition We say that PL(κ) is (*)-compact if every set of formulas F which is κ-satisfiable, and (*) this satisfiability is witnessed by a κ-tree of evaluations, is satisfiable.

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Some compactness principles

Compactness of successor cardinals

Note that if κ is inaccessible, the condition (*) in the Definition of compactness for PL(κ) is redundant, i.e. compactness and (*)-compactness are equivalent for ω and all inaccessible κ ≥ ω.

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Some compactness principles

Compactness of successor cardinals

Note that if κ is inaccessible, the condition (*) in the Definition of compactness for PL(κ) is redundant, i.e. compactness and (*)-compactness are equivalent for ω and all inaccessible κ ≥ ω. Also note that compactness of PL(κ) implies inaccessibility of κ, and therefore (*)-compactness is strictly weaker.

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Some compactness principles

Compactness of successor cardinals

Note that if κ is inaccessible, the condition (*) in the Definition of compactness for PL(κ) is redundant, i.e. compactness and (*)-compactness are equivalent for ω and all inaccessible κ ≥ ω. Also note that compactness of PL(κ) implies inaccessibility of κ, and therefore (*)-compactness is strictly weaker. We obtain the following generalization:

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Some compactness principles

Compactness of successor cardinals

Note that if κ is inaccessible, the condition (*) in the Definition of compactness for PL(κ) is redundant, i.e. compactness and (*)-compactness are equivalent for ω and all inaccessible κ ≥ ω. Also note that compactness of PL(κ) implies inaccessibility of κ, and therefore (*)-compactness is strictly weaker. We obtain the following generalization: Theorem Let κ ≥ ω be a regular cardinal. Then the following are equivalent: PL(κ) is (*)-compact. Every κ-tree has a cofinal branch, TP(κ).

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Some compactness principles

Some properties of (*)-compactness

Provably in ZFC, PL(ω1) is not (*)-compact. Counterexamples of evaluations which form an ω1-tree without a cofinal branch are called Aronszajn trees.

2See my webpage for more details.

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Some compactness principles

Some properties of (*)-compactness

Provably in ZFC, PL(ω1) is not (*)-compact. Counterexamples of evaluations which form an ω1-tree without a cofinal branch are called Aronszajn trees. (*)-compactness has an intriguing effect on the size of powersets: for instance if PL(ω2) is (*)-compact, then ¬CH.

2See my webpage for more details.

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Some compactness principles

Some properties of (*)-compactness

Provably in ZFC, PL(ω1) is not (*)-compact. Counterexamples of evaluations which form an ω1-tree without a cofinal branch are called Aronszajn trees. (*)-compactness has an intriguing effect on the size of powersets: for instance if PL(ω2) is (*)-compact, then ¬CH. In general, instances of GCH imply that PL(κ) is not (*)-compact for many κ. With some large cardinal assumptions, PL(ωn) can be (*)-compact for every 1 < n < ω simultaneously.

2See my webpage for more details.

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Some compactness principles

Some properties of (*)-compactness

Provably in ZFC, PL(ω1) is not (*)-compact. Counterexamples of evaluations which form an ω1-tree without a cofinal branch are called Aronszajn trees. (*)-compactness has an intriguing effect on the size of powersets: for instance if PL(ω2) is (*)-compact, then ¬CH. In general, instances of GCH imply that PL(κ) is not (*)-compact for many κ. With some large cardinal assumptions, PL(ωn) can be (*)-compact for every 1 < n < ω simultaneously. Consistently PL(ℵω+2) can be (*)-compact, with 2ℵω being equal to ℵω+n for any prescribed 2 ≤ n < ω, etc.2

2See my webpage for more details.

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Some compactness principles

An open question

However the following question remains open and drives the current research in this area:

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Some compactness principles

An open question

However the following question remains open and drives the current research in this area: Question Does ZFC prove that there is at least one regular κ > ω1 such that PL(κ) is not (*)-compact?

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Some compactness principles

An open question

However the following question remains open and drives the current research in this area: Question Does ZFC prove that there is at least one regular κ > ω1 such that PL(κ) is not (*)-compact? A negative answer would mean that ω is the unique infinite cardinal κ which has the effect of making PL(κ+) provably (*)-incompact, i.e. that it is consistent that PL(κ) is (*)-compact for every regular κ except ω1.

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Some compactness principles

An open question

However the following question remains open and drives the current research in this area: Question Does ZFC prove that there is at least one regular κ > ω1 such that PL(κ) is not (*)-compact? A negative answer would mean that ω is the unique infinite cardinal κ which has the effect of making PL(κ+) provably (*)-incompact, i.e. that it is consistent that PL(κ) is (*)-compact for every regular κ except ω1. Note that, as it often happens, these questions have been studied directly through the combinatorial properties of trees, TP(κ), with the logical context being only implicit in the discussion.

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Some compactness principles

LS arguments

Assume that a regular cardinal κ > ω is a domain of a structure, and let us consider substructures with domains formed by ordinals α < κ. A natural question is to ask “how many” such substructures there are. The answer is an easy application of an LS argument, but leads to the important definition of a closed unbounded set (club):

• R. Honzik

Some compactness principles

LS arguments

Assume that a regular cardinal κ > ω is a domain of a structure, and let us consider substructures with domains formed by ordinals α < κ. A natural question is to ask “how many” such substructures there are. The answer is an easy application of an LS argument, but leads to the important definition of a closed unbounded set (club): we say that C ⊆ κ is unbounded if for every α < κ there is β ∈ C with α ≤ β; we say that C is closed if for every limit α < κ, whenever C ∩ α is unbounded in α, then α ∈ C.

• R. Honzik

Some compactness principles

LS arguments

Assume that a regular cardinal κ > ω is a domain of a structure, and let us consider substructures with domains formed by ordinals α < κ. A natural question is to ask “how many” such substructures there are. The answer is an easy application of an LS argument, but leads to the important definition of a closed unbounded set (club): we say that C ⊆ κ is unbounded if for every α < κ there is β ∈ C with α ≤ β; we say that C is closed if for every limit α < κ, whenever C ∩ α is unbounded in α, then α ∈ C. Theorem (L¨

• wenheim-Skolem)

Let f : κ → κ be a function and consider the structure (κ, f ) where κ > ω is regular. Then the set of all α such that (α, f |α) is a substructure of (κ, f ) is a club.

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Some compactness principles

LS arguments

Let us note the following:

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Some compactness principles

LS arguments

Let us note the following: A typical example of a club subset of κ is the set of all limit

• rdinals α < κ.
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Some compactness principles

LS arguments

Let us note the following: A typical example of a club subset of κ is the set of all limit

• rdinals α < κ.

One can show that club sets generate a filter, which we call the closed unbounded filter at κ, F(κ). This filter is uniform, κ-complete, and normal. With AC, it is not an ultrafilter.

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Some compactness principles

LS arguments

Let us note the following: A typical example of a club subset of κ is the set of all limit

• rdinals α < κ.

One can show that club sets generate a filter, which we call the closed unbounded filter at κ, F(κ). This filter is uniform, κ-complete, and normal. With AC, it is not an ultrafilter. Notice that κ is required to be uncountable: a club subset of ω is just an unbounded set, and such sets do not generate a filter (it is easy to find two disjoint unbounded subsets of ω).

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Some compactness principles

LS arguments

Let us note the following: A typical example of a club subset of κ is the set of all limit

• rdinals α < κ.

One can show that club sets generate a filter, which we call the closed unbounded filter at κ, F(κ). This filter is uniform, κ-complete, and normal. With AC, it is not an ultrafilter. Notice that κ is required to be uncountable: a club subset of ω is just an unbounded set, and such sets do not generate a filter (it is easy to find two disjoint unbounded subsets of ω). The κ-completeness implies that the substructures can be defined with respect to < κ-many functions fα simultaneously.

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Some compactness principles

Stationary sets

Definition The dual ideal to the club filter F(κ) at κ is called the non-stationary ideal and is denoted NS(κ).

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Some compactness principles

Stationary sets

Definition The dual ideal to the club filter F(κ) at κ is called the non-stationary ideal and is denoted NS(κ). A subset of κ is called stationary if it is not in NS(κ). Let us denote stationary subsets of κ by S(κ).

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Some compactness principles

Stationary sets

Definition The dual ideal to the club filter F(κ) at κ is called the non-stationary ideal and is denoted NS(κ). A subset of κ is called stationary if it is not in NS(κ). Let us denote stationary subsets of κ by S(κ). The following summarizes our notation:

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Some compactness principles

Stationary sets

Definition The dual ideal to the club filter F(κ) at κ is called the non-stationary ideal and is denoted NS(κ). A subset of κ is called stationary if it is not in NS(κ). Let us denote stationary subsets of κ by S(κ). The following summarizes our notation: P(κ) = S(κ) ∪ NS(κ), S(κ) ∩ NS(κ) = ∅, F(κ) S(κ).

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Some compactness principles

Stationary sets

An equivalent characterization of stationarity is that S ⊆ κ is stationary if it meets every club set C: it follows that even though S may not be closed (but is always unbounded), S does contain at least one α such that (α, f |α) is a substructure of (κ, f ) for any fixed f .

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Some compactness principles

Stationary sets

An equivalent characterization of stationarity is that S ⊆ κ is stationary if it meets every club set C: it follows that even though S may not be closed (but is always unbounded), S does contain at least one α such that (α, f |α) is a substructure of (κ, f ) for any fixed f . (In fact, it is easy to show that S must contain unboundedly many such α.)

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Some compactness principles

Stationary sets

An equivalent characterization of stationarity is that S ⊆ κ is stationary if it meets every club set C: it follows that even though S may not be closed (but is always unbounded), S does contain at least one α such that (α, f |α) is a substructure of (κ, f ) for any fixed f . (In fact, it is easy to show that S must contain unboundedly many such α.) This property makes stationary sets very useful.

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Some compactness principles

Stationary sets

An equivalent characterization of stationarity is that S ⊆ κ is stationary if it meets every club set C: it follows that even though S may not be closed (but is always unbounded), S does contain at least one α such that (α, f |α) is a substructure of (κ, f ) for any fixed f . (In fact, it is easy to show that S must contain unboundedly many such α.) This property makes stationary sets very useful. Unlike club sets, stationary sets may be disjoint:

• R. Honzik

Some compactness principles

Stationary sets

An equivalent characterization of stationarity is that S ⊆ κ is stationary if it meets every club set C: it follows that even though S may not be closed (but is always unbounded), S does contain at least one α such that (α, f |α) is a substructure of (κ, f ) for any fixed f . (In fact, it is easy to show that S must contain unboundedly many such α.) This property makes stationary sets very useful. Unlike club sets, stationary sets may be disjoint: Theorem (Solovay) Let κ > ω be a regular cardinal. Then any stationary set S ⊆ κ can be partitioned into κ-many disjoint stationary sets.

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Some compactness principles

Reflection of club sets

Let κ > ω1 be regular and let C ⊆ κ be club. Let us denote by Lim(C) the set of limit points of C: i.e. the set of all limit ordinals α < κ such that C ∩ α is unbounded in α.

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Some compactness principles

Reflection of club sets

Let κ > ω1 be regular and let C ⊆ κ be club. Let us denote by Lim(C) the set of limit points of C: i.e. the set of all limit ordinals α < κ such that C ∩ α is unbounded in α. It is easy to check that Lim(C) ⊆ C is also a club set.

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Some compactness principles

Reflection of club sets

Let κ > ω1 be regular and let C ⊆ κ be club. Let us denote by Lim(C) the set of limit points of C: i.e. the set of all limit ordinals α < κ such that C ∩ α is unbounded in α. It is easy to check that Lim(C) ⊆ C is also a club set. Observation If C is a club subset of κ, then for every α ∈ Lim(C) with uncountable cofinality, α ∩ C is a club subset of α.

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Some compactness principles

Reflection of club sets

Let κ > ω1 be regular and let C ⊆ κ be club. Let us denote by Lim(C) the set of limit points of C: i.e. the set of all limit ordinals α < κ such that C ∩ α is unbounded in α. It is easy to check that Lim(C) ⊆ C is also a club set. Observation If C is a club subset of κ, then for every α ∈ Lim(C) with uncountable cofinality, α ∩ C is a club subset of α. We say that C reflects at α < κ.

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Some compactness principles

Stationary reflection

We can ask whether the reflection enjoyed by the club sets carries

• ver to stationary sets:
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Some compactness principles

Stationary reflection

We can ask whether the reflection enjoyed by the club sets carries

• ver to stationary sets:

Question: Suppose κ > ω1 is regular. Does every stationary set S ⊆ κ reflect at a point of uncountable cofinality?

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Some compactness principles

Stationary reflection

We can ask whether the reflection enjoyed by the club sets carries

• ver to stationary sets:

Question: Suppose κ > ω1 is regular. Does every stationary set S ⊆ κ reflect at a point of uncountable cofinality? It is easy to come up with a counterexample for a successor κ = µ+, so we should be more careful which stationary sets should reflect (inaccessible κ do not have such counterexamples).

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Some compactness principles

Stationary reflection

We can ask whether the reflection enjoyed by the club sets carries

• ver to stationary sets:

Question: Suppose κ > ω1 is regular. Does every stationary set S ⊆ κ reflect at a point of uncountable cofinality? It is easy to come up with a counterexample for a successor κ = µ+, so we should be more careful which stationary sets should reflect (inaccessible κ do not have such counterexamples). [Details: Let κ = ω2 for simplicity and let E be the set of all

• rdinals α < ω2 with cofinality ω1.
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Some compactness principles

Stationary reflection

We can ask whether the reflection enjoyed by the club sets carries

• ver to stationary sets:

Question: Suppose κ > ω1 is regular. Does every stationary set S ⊆ κ reflect at a point of uncountable cofinality? It is easy to come up with a counterexample for a successor κ = µ+, so we should be more careful which stationary sets should reflect (inaccessible κ do not have such counterexamples). [Details: Let κ = ω2 for simplicity and let E be the set of all

• rdinals α < ω2 with cofinality ω1. It is easy to find for each α in

E a club Cα ⊆ α which is composed only of ordinals with countable cofinality.

• R. Honzik

Some compactness principles

Stationary reflection

We can ask whether the reflection enjoyed by the club sets carries

• ver to stationary sets:

Question: Suppose κ > ω1 is regular. Does every stationary set S ⊆ κ reflect at a point of uncountable cofinality? It is easy to come up with a counterexample for a successor κ = µ+, so we should be more careful which stationary sets should reflect (inaccessible κ do not have such counterexamples). [Details: Let κ = ω2 for simplicity and let E be the set of all

• rdinals α < ω2 with cofinality ω1. It is easy to find for each α in

E a club Cα ⊆ α which is composed only of ordinals with countable cofinality. It follows that E ∩ α is disjoint from Cα for every α ∈ E, and therefore cannot be stationary.

• R. Honzik

Some compactness principles

Stationary reflection

We can ask whether the reflection enjoyed by the club sets carries

• ver to stationary sets:

Question: Suppose κ > ω1 is regular. Does every stationary set S ⊆ κ reflect at a point of uncountable cofinality? It is easy to come up with a counterexample for a successor κ = µ+, so we should be more careful which stationary sets should reflect (inaccessible κ do not have such counterexamples). [Details: Let κ = ω2 for simplicity and let E be the set of all

• rdinals α < ω2 with cofinality ω1. It is easy to find for each α in

E a club Cα ⊆ α which is composed only of ordinals with countable cofinality. It follows that E ∩ α is disjoint from Cα for every α ∈ E, and therefore cannot be stationary. In other words, E does not reflect at any ordinal of uncountable cofinality.]

• R. Honzik

Some compactness principles

Stationary reflection

Let us state our next definition a bit vaguely:

• R. Honzik

Some compactness principles

Stationary reflection

Let us state our next definition a bit vaguely: Definition We say that a regular cardinal κ > ω1 satisfies the stationary reflection, SR(κ), if (*)-every stationary subset of κ reflects at a point of uncountable cofinality,

• R. Honzik

Some compactness principles

Stationary reflection

Let us state our next definition a bit vaguely: Definition We say that a regular cardinal κ > ω1 satisfies the stationary reflection, SR(κ), if (*)-every stationary subset of κ reflects at a point of uncountable cofinality, where “(*)-every” means “except for those which cannot provably reflect” (such as E from the previous slide).

• R. Honzik

Some compactness principles

Stationary reflection

Let us state our next definition a bit vaguely: Definition We say that a regular cardinal κ > ω1 satisfies the stationary reflection, SR(κ), if (*)-every stationary subset of κ reflects at a point of uncountable cofinality, where “(*)-every” means “except for those which cannot provably reflect” (such as E from the previous slide). Note that if κ is inaccessible, then “(*)-every” is the same as “every”.

• R. Honzik

Some compactness principles

Tree property + stationary reflection

While the notions of the tree property TP(κ) and stationary reflection SR(κ) emerge from different considerations about logic, they are related in the following sense:

• R. Honzik

Some compactness principles

Tree property + stationary reflection

While the notions of the tree property TP(κ) and stationary reflection SR(κ) emerge from different considerations about logic, they are related in the following sense: Theorem Suppose κ is inaccessible, then TP(κ) → SR(κ).

• R. Honzik

Some compactness principles

Tree property + stationary reflection

While the notions of the tree property TP(κ) and stationary reflection SR(κ) emerge from different considerations about logic, they are related in the following sense: Theorem Suppose κ is inaccessible, then TP(κ) → SR(κ). The converse is not generally true, but one can formulate stationary reflection in a stronger way to obtain the equivalence.

• R. Honzik

Some compactness principles

Tree property + stationary reflection

While the notions of the tree property TP(κ) and stationary reflection SR(κ) emerge from different considerations about logic, they are related in the following sense: Theorem Suppose κ is inaccessible, then TP(κ) → SR(κ). The converse is not generally true, but one can formulate stationary reflection in a stronger way to obtain the equivalence. It has been shown recently that for a successor κ, TP(κ) and SR(κ) are not implying each other.

• R. Honzik

Some compactness principles

Research and methods

Let us end the talk with references and notes for further research:

• R. Honzik

Some compactness principles

Research and methods

Let us end the talk with references and notes for further research: It has been investigated whether compactness principles such as TP(κ) and SR(κ) influence other combinatorial properties

• f sets, in particular the values of the continuum function

which maps a cardinal µ to its size 2µ (the case when κ is the successor or double successor of a singular cardinal is the most challenging). See my webpage for more references and results for this topic.

• R. Honzik

Some compactness principles

Research and methods

Let us end the talk with references and notes for further research: It has been investigated whether compactness principles such as TP(κ) and SR(κ) influence other combinatorial properties

• f sets, in particular the values of the continuum function

which maps a cardinal µ to its size 2µ (the case when κ is the successor or double successor of a singular cardinal is the most challenging). See my webpage for more references and results for this topic. The methods for obtaining successor cardinals µ+ with TP(µ+) or SR(µ+) involve the use of forcing, and usually start with a weakly compact cardinal κ, which is then collapsed to become µ+. For instance TP(ω2) + SR(ω2) can be obtained in this way.

• R. Honzik

Some compactness principles

A driving question

Let us consider the following (vague) question:

3For instance TP(ω2) implies that ω2 is weakly compact in L ⊆ V .

• R. Honzik

Some compactness principles

A driving question

Let us consider the following (vague) question: Question Is it consistent that the universe V is “globally compact” in the sense that every cardinal κ ≥ ω which can be in principle compact (in any desired sense) really is compact? For instance can TP(κ) hold for every regular κ > ω1? Can SR(κ) hold for every such κ, and can they hold together?

3For instance TP(ω2) implies that ω2 is weakly compact in L ⊆ V .

• R. Honzik

Some compactness principles

A driving question

Let us consider the following (vague) question: Question Is it consistent that the universe V is “globally compact” in the sense that every cardinal κ ≥ ω which can be in principle compact (in any desired sense) really is compact? For instance can TP(κ) hold for every regular κ > ω1? Can SR(κ) hold for every such κ, and can they hold together? Let us finish by noting that if the answer is positive, every cardinal in V can be viewed as an extremely large (in particular inaccessible) cardinal in a smaller subuniverse M ⊆ V .3

3For instance TP(ω2) implies that ω2 is weakly compact in L ⊆ V .

• R. Honzik

Some compactness principles