Small ultrafilter number and some compactness principles S arka - PowerPoint PPT Presentation

Small ultrafilter number and some compactness principles ˇ S´ arka Stejskalov´ a Department of Logic, Charles University Institute of Mathematics, Czech Academy of Sciences logika.ff.cuni.cz/sarka Bristol February 3-4, 2020 ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

We will show that the following three properties are consistent together for a strong limit singular κ of countable or uncountable cofinality: 2 κ > κ + , u ( κ ) = κ + , κ ++ is compact in a prescribed sense. 1 1 The tree property, TP, stationary reflection, SR, and non-approachability ¬ AP. ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

We will show that the following three properties are consistent together for a strong limit singular κ of countable or uncountable cofinality: 2 κ > κ + , u ( κ ) = κ + , κ ++ is compact in a prescribed sense. 1 This result extends the existing results which study the interplay between compactness and the continuum function by including one more cardinal invariant: the ultrafilter number u ( κ ). 1 The tree property, TP, stationary reflection, SR, and non-approachability ¬ AP. ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

Ultrafilter number We say that an ultrafilter U on an infinite cardinal κ is uniform if every X ∈ U has size κ . ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

Ultrafilter number We say that an ultrafilter U on an infinite cardinal κ is uniform if every X ∈ U has size κ . Definition u ( κ ) is the least cardinal α such that there exists a base B of a uniform ultrafilter U on κ of size α , where B is a base of U if for every X ∈ U there is Y ∈ B with Y ⊆ X . ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

Ultrafilter number We say that an ultrafilter U on an infinite cardinal κ is uniform if every X ∈ U has size κ . Definition u ( κ ) is the least cardinal α such that there exists a base B of a uniform ultrafilter U on κ of size α , where B is a base of U if for every X ∈ U there is Y ∈ B with Y ⊆ X . For all infinite κ , u ( κ ) is always at least κ + . ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

Ultrafilter number at ω Let us note that for κ = ω , it is known that u ( ω ) = ω 1 is consistent with 2 ω large: for instance the product or iteration of the Sacks forcing of length ω 2 , or an iteration of length ω 1 over a model of ¬ CH of Mathias forcing, achieves this configuration. ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

Ultrafilter number at ω Let us note that for κ = ω , it is known that u ( ω ) = ω 1 is consistent with 2 ω large: for instance the product or iteration of the Sacks forcing of length ω 2 , or an iteration of length ω 1 over a model of ¬ CH of Mathias forcing, achieves this configuration. The iteration of Sacks forcing also forces TP( ω 2 ) and SR( ω 2 ) if ω 2 used to be weakly compact in the ground model. ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

Ultrafilter number at ω Let us note that for κ = ω , it is known that u ( ω ) = ω 1 is consistent with 2 ω large: for instance the product or iteration of the Sacks forcing of length ω 2 , or an iteration of length ω 1 over a model of ¬ CH of Mathias forcing, achieves this configuration. The iteration of Sacks forcing also forces TP( ω 2 ) and SR( ω 2 ) if ω 2 used to be weakly compact in the ground model. The situation for κ > ω is less understood. ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

Ultrafilter number at κ > ω Current research gives more information about small u ( κ ) for limit cardinals κ : ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

Ultrafilter number at κ > ω Current research gives more information about small u ( κ ) for limit cardinals κ : New results have appeared recently for a strong limit singular κ or its successor . The fact that κ is a limit of < κ many cardinals gives a space to construct ultrafilters with small bases (using pcf-style arguments). ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

Ultrafilter number at κ > ω Current research gives more information about small u ( κ ) for limit cardinals κ : New results have appeared recently for a strong limit singular κ or its successor . The fact that κ is a limit of < κ many cardinals gives a space to construct ultrafilters with small bases (using pcf-style arguments). Some techniques are available for weakly or strongly inaccessible cardinals. ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

Ultrafilter number at κ > ω Current research gives more information about small u ( κ ) for limit cardinals κ : New results have appeared recently for a strong limit singular κ or its successor . The fact that κ is a limit of < κ many cardinals gives a space to construct ultrafilters with small bases (using pcf-style arguments). Some techniques are available for weakly or strongly inaccessible cardinals. The following, however, is still open: Question Is u ( κ ) < 2 κ consistent for a successor of a regular cardinal? ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

“Compactness at κ ++ ” can mean many things, we will in the talk focus on three: the tree property, stationary reflection, the failure of approachability. ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

The tree property Definition Let λ be a regular cardinal. We say that the tree property holds at λ , and we write TP( λ ), if every λ -tree has a cofinal branch. ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

The tree property Definition Let λ be a regular cardinal. We say that the tree property holds at λ , and we write TP( λ ), if every λ -tree has a cofinal branch. Note that with inaccessibility of λ , TP( λ ) is equivalent to λ being weakly compact. Over L , TP( λ ) is equivalent to λ being weakly compact. ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

The tree property TP( ℵ 0 ) and ¬ TP( ℵ 1 ). ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

The tree property TP( ℵ 0 ) and ¬ TP( ℵ 1 ). (Specker) If κ <κ = κ then there exists a κ + -Aronszajn tree. Therefore ¬ TP( κ + ). ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

The tree property TP( ℵ 0 ) and ¬ TP( ℵ 1 ). (Specker) If κ <κ = κ then there exists a κ + -Aronszajn tree. Therefore ¬ TP( κ + ). If GCH then ¬ TP( κ ++ ) for all κ ≥ ω . ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

The tree property TP( ℵ 0 ) and ¬ TP( ℵ 1 ). (Specker) If κ <κ = κ then there exists a κ + -Aronszajn tree. Therefore ¬ TP( κ + ). If GCH then ¬ TP( κ ++ ) for all κ ≥ ω . TP( κ ++ ) then 2 κ > κ + . ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

The tree property TP( ℵ 0 ) and ¬ TP( ℵ 1 ). (Specker) If κ <κ = κ then there exists a κ + -Aronszajn tree. Therefore ¬ TP( κ + ). If GCH then ¬ TP( κ ++ ) for all κ ≥ ω . TP( κ ++ ) then 2 κ > κ + . It follows that the tree property has a non-trivial effect on the continuum function. It is of interest to study the extent of this effect to cardinal invariants. ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

Stationary reflection Definition Let λ be a cardinal of the form λ = ν + for some regular cardinal ν . We say that the stationary reflection holds at λ , and write SR( λ ), if every stationary subset S ⊆ λ ∩ cof( < ν ) reflects at a point of cofinality ν ; i.e. there is α < λ of cofinality ν such that α ∩ S is stationary in α . ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles

Stationary reflection Definition Let λ be a cardinal of the form λ = ν + for some regular cardinal ν . We say that the stationary reflection holds at λ , and write SR( λ ), if every stationary subset S ⊆ λ ∩ cof( < ν ) reflects at a point of cofinality ν ; i.e. there is α < λ of cofinality ν such that α ∩ S is stationary in α . Stationary subsets of λ ∩ cof( ν ) never reflect. Also note that over L , if every stationary subset of λ reflects, then λ is weakly compact. ˇ S´ arka Stejskalov´ a Small ultrafilter number and some compactness principles