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¨ owenheim-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¨ owenheim-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 . 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 ordinal-type of the wellordering of the predecessors of t . 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 ordinal-type of the wellordering of the predecessors of t . For every α , let Lev T ( 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 ordinal-type of the wellordering of the predecessors of t . For every α , let Lev T ( 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 Lev T ( 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 ordinal-type of the wellordering of the predecessors of t . For every α , let Lev T ( 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 Lev T ( α ) is empty. T is called a κ -tree , where κ is a cardinal, if ht ( T ) = κ , and for all α < ht ( T ), | Lev T ( α ) | < κ . R. Honzik 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. R. Honzik 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 <α 2 be the set of all functions t : β → 2, for β < α . The pair T = ( <α 2 , ⊆ ) 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. 2 µ < α for every µ < α ). R. Honzik 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 <α 2 be the set of all functions t : β → 2, for β < α . The pair T = ( <α 2 , ⊆ ) 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. 2 µ < α for every µ < α ). In general, ( <α β, ⊆ ) is a tree of height α in which each nodes splits into | β | -many successors. R. Honzik 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 <α 2 be the set of all functions t : β → 2, for β < α . The pair T = ( <α 2 , ⊆ ) 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. 2 µ < α for every µ < α ). In general, ( <α β, ⊆ ) is a tree of height α in which each nodes splits into | β | -many successors. We say that T ⊆ <α β together with the inclusion relation is a tree if it is closed under initial segments. R. Honzik 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 ∩ Lev T ( α ) is non-empty for every α < ht ( T ). R. Honzik 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 ∩ Lev T ( α ) is non-empty for every α < ht ( T ). The tree T = ( <ω 2 , ⊆ ) has 2 ω -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 ω 2 the Cantor space ). Notice that T is an ω -tree since all its levels are finite. R. Honzik 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 ∩ Lev T ( α ) is non-empty for every α < ht ( T ). The tree T = ( <ω 2 , ⊆ ) has 2 ω -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 ω 2 the Cantor space ). Notice that T is an ω -tree since all its levels are finite. The tree T = ( <ω ω, ⊆ ) is not an ω -tree and its (all cofinal) branches are elements of ω ω , which we call the Baire space . R. Honzik Some compactness principles

Trees: branches For every k < ω , let A k denote the set of all pairs ( m , k ) such that m < k and consider the set T = � k <ω A k . 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. R. Honzik Some compactness principles

Trees: branches For every k < ω , let A k denote the set of all pairs ( m , k ) such that m < k and consider the set T = � k <ω A k . 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 ω 1 which has no cofinal branch. Note that T has levels of size 2 ω , and so T is not an ω 1 -tree. R. Honzik Some compactness principles

Question Question: If κ ≥ ω is a regular cardinal, does every κ -tree have a cofinal branch? R. Honzik 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 ). R. Honzik Some compactness principles

ω -trees Let us first deal with κ = ω . R. Honzik Some compactness principles

ω -trees Let us first deal with κ = ω . Theorem (K¨ onig) Every ω -tree has a cofinal branch. R. Honzik Some compactness principles