Lower Bounds for the Perfect Subtree Property at Weakly Compact Cardinals Sandra M¨ uller Universit¨ at Wien September 23, 2019 Joint with Yair Hayut 15th International Luminy Workshop in Set Theory Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 1
κ A N - o T F B n r i r e - e B a s f d e l t o r n o o s w o m d L u a m o n n S W e d w s d e t t s e o h a i U r o e c l e p B d d κ M p i o n - e T P o u u r r c S n e a s d P e r e d s i n a l s Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 2
κ -Trees Definition Let κ be a regular cardinal. A tree T of height κ is called a normal κ -tree if each level of T has size <κ , each level has at least one split, for every limit ordinal α < κ and every branch up to α there is at most one least upper bound in T , and for every t ∈ T and α < κ above the height of t , there is some t ′ of level α in T such that t < T t ′ . Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 3
The Branch Spectrum Definition Let κ be a regular cardinal. The Branch Spectrum of κ is the set S κ = {| [ T ] | | T is a normal κ -tree } . Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 4
The Branch Spectrum Definition Let κ be a regular cardinal. The Branch Spectrum of κ is the set S κ = {| [ T ] | | T is a normal κ -tree } . Examples S ω = {ℵ 0 , 2 ℵ 0 } . Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 4
The Branch Spectrum Definition Let κ be a regular cardinal. The Branch Spectrum of κ is the set S κ = {| [ T ] | | T is a normal κ -tree } . Examples S ω = {ℵ 0 , 2 ℵ 0 } . For κ > ω , if there are no κ -Kurepa trees, then κ + / ∈ S κ . Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 4
The Branch Spectrum Definition Let κ be a regular cardinal. The Branch Spectrum of κ is the set S κ = {| [ T ] | | T is a normal κ -tree } . Examples S ω = {ℵ 0 , 2 ℵ 0 } . For κ > ω , if there are no κ -Kurepa trees, then κ + / ∈ S κ . For κ > ℵ 1 , if the tree property holds at κ , then min( S κ ) = κ . Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 4
Upper Bounds Let κ > ℵ 1 . Branch Spectrum Upper bound κ + / ∈ S κ inaccessible cardinal Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 5
Upper Bounds Let κ > ℵ 1 . Branch Spectrum Upper bound κ + / ∈ S κ inaccessible cardinal min( S κ ) = κ weakly compact cardinal Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 5
Upper Bounds Let κ > ℵ 1 . Branch Spectrum Upper bound κ + / ∈ S κ inaccessible cardinal min( S κ ) = κ weakly compact cardinal κ + / ∈ S κ and min( S κ ) = κ ? Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 5
Upper Bounds Let κ > ℵ 1 . Branch Spectrum Upper bound κ + / ∈ S κ inaccessible cardinal min( S κ ) = κ weakly compact cardinal κ + / ∈ S κ and min( S κ ) = κ ? The following gives an upper bound. Proposition Let κ be <µ -supercompact, where µ is strongly inaccessible. Then, there is a forcing extension in which κ is weakly compact, S κ = { κ, κ ++ } . Proof idea: Consider Col( κ, <µ ) × Add( κ, µ + ) . Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 5
Upper Bounds Let κ > ℵ 1 . Branch Spectrum Upper bound κ + / ∈ S κ inaccessible cardinal min( S κ ) = κ weakly compact cardinal κ + / ∈ S κ and min( S κ ) = κ ? The following gives an upper bound. Proposition Let κ be <µ -supercompact, where µ is strongly inaccessible. Then, there is a forcing extension in which κ is weakly compact, S κ = { κ, κ ++ } . Proof idea: Consider Col( κ, <µ ) × Add( κ, µ + ) . Question Is this optimal? Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 5
κ A N - o T F B n r i r e - e B a s f d e l t o r n o o s w o m d L u a m o n n S W e d w s d e t t s e o h a i U r o e c l e p B d d κ M p i o n - e T P o u u r r c S n e a s d P e r e d s i n a l s Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 6
A First Lower Bound and Sealed Trees If for some weakly compact cardinal κ , κ + / ∈ S κ then 0 # exists: Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 7
A First Lower Bound and Sealed Trees If for some weakly compact cardinal κ , κ + / ∈ S κ then 0 # exists: Fact (essentially Solovay) If 0 # does not exists then every weakly compact cardinal carries a tree with κ + many branches. Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 7
A First Lower Bound and Sealed Trees If for some weakly compact cardinal κ , κ + / ∈ S κ then 0 # exists: Fact (essentially Solovay) If 0 # does not exists then every weakly compact cardinal carries a tree with κ + many branches. The tree actually has the following stronger property. Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 7
A First Lower Bound and Sealed Trees If for some weakly compact cardinal κ , κ + / ∈ S κ then 0 # exists: Fact (essentially Solovay) If 0 # does not exists then every weakly compact cardinal carries a tree with κ + many branches. The tree actually has the following stronger property. Definition Let κ be a regular cardinal. A normal tree T of height κ is strongly sealed if the set of branches of T cannot be modified by set forcing that forces cf( κ ) > ω . Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 7
A First Lower Bound and Sealed Trees The tree actually has the following stronger property. Definition Let κ be a regular cardinal. A normal tree T of height κ is strongly sealed if the set of branches of T cannot be modified by set forcing that forces cf( κ ) > ω . Strongly sealed trees with κ many branches exist in ZFC : Take T ⊆ 2 <κ to be the tree of all x such that { α ∈ dom( x ) | x ( α ) = 1 } is finite. Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 7
A First Lower Bound and Sealed Trees The tree actually has the following stronger property. Definition Let κ be a regular cardinal. A normal tree T of height κ is strongly sealed if the set of branches of T cannot be modified by set forcing that forces cf( κ ) > ω . Strongly sealed trees with κ many branches exist in ZFC : Take T ⊆ 2 <κ to be the tree of all x such that { α ∈ dom( x ) | x ( α ) = 1 } is finite. Question How about strongly sealed κ -trees with at least κ + many branches? Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 7
κ A N - o T F B n r i r e - e B a s f d e l t o r n o o s w o m d L u a m o n n S W e d w s d e t t s e o h a i U r o e c l e p B d d κ M p i o n - e T P o u u r r c S n e a s d P e r e d s i n a l s Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 8
A Sealed Tree in K Theorem (Hayut, M.) Let us assume that there is no inner model with a Woodin cardinal. Then for every regular cardinal κ , there is a strongly sealed κ -tree with exactly ( κ + ) K many branches. In particular, if κ is weakly compact, then there is a strongly sealed tree on κ with κ + many branches. Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 9
A Sealed Tree in K Theorem (Hayut, M.) Let us assume that there is no inner model with a Woodin cardinal. Then for every regular cardinal κ , there is a strongly sealed κ -tree with exactly ( κ + ) K many branches. In particular, if κ is weakly compact, then there is a strongly sealed tree on κ with κ + many branches. Proof idea: Construct a κ -tree T in K with | [ T ] | ≥ ( κ + ) K . Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 9
Construction of the Tree Let T ( K κ + ) be the following tree: Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 9
Construction of the Tree Let T ( K κ + ) be the following tree: Nodes: � ¯ x � , where ¯ M = trcl(Hull K κ + ( ρ ∪ { x } )) for some ρ < κ , M, ¯ x ∈ K κ + ∩ κ 2 and x collapses to ¯ x . Sandra M¨ uller (Universit¨ at Wien) Perfect Subtree Property at Weakly Compacts September 23, 2019 9
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