A Journey Through the World of Mice and Games Projective and Beyond - PowerPoint PPT Presentation

A Journey Through the World of Mice and Games Projective and Beyond Sandra Uhlenbrock June 13th, 2016 Young Set Theory Workshop Copenhagen Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 1 / 29

Descriptive Set Theory Inner Model Theory Beyond the Projective Hierarchy Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 2 / 29

Descriptive Set Theory Inner Model Theory Beyond the Projective Hierarchy Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 3 / 29

Games in Set Theory Definition (Gale/Stewart 1953) Let A ⊂ 2 N . With G ( A ) we denote the following game I i 0 i 2 . . . for i n ∈ { 0 , 1 } and n ∈ N . II i 1 i 3 . . . We say player I wins the game iff ( i n ) n ∈ N ∈ A . Otherwise player II wins. We say G ( A ) (or A itself) is determined iff one of the players has a winning strategy (in the obvious sense). Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 4 / 29

Which games are determined? Theorem (Gale/Stewart, 1953) (AC) Let A ⊂ 2 N be open or closed. Then G ( A ) is determined. Theorem (Gale/Stewart, 1953) Assuming AC there is a set of reals which is not determined. Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 5 / 29

The Projective Hierarchy Let B be the collection of all Borel sets of reals. Then we define the projective hierarchy as follows. Σ 1 1 = analytic sets, i.e. projections of Borel sets , Π 1 n = complements of sets in Σ 1 n , Σ 1 n +1 = projections of sets in Π 1 n . A set is projective if it is in Σ 1 n (or Π 1 n ) for some n . Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 6 / 29

Determinacy for Different Sets of Reals Theorem (Martin, 1975) Assume ZFC . Then every Borel set of reals is determined. Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 7 / 29

Determinacy for Different Sets of Reals Theorem (Martin, 1975) Assume ZFC . Then every Borel set of reals is determined. Determinacy for all projective sets of reals is not provable in ZFC alone. Theorem (Martin/Steel, 1985) Assume ZFC and there are n Woodin cardinals with a measurable cardinal above them all. Then every Σ 1 n +1 set is determined. Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 7 / 29

Descriptive Set Theory Inner Model Theory Beyond the Projective Hierarchy Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 8 / 29

Inner Model Theory The main goal of inner model theory is to construct L -like models, which we call mice, for stronger and stronger large cardinals. Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 9 / 29

G¨ odel’s constructible universe L Definition Let E be a set or a proper class. Let J 0 [ E ] = ∅ J α +1 [ E ] = rud E ( J α [ E ] ∪ { J α [ E ] } ) � J λ [ E ] = J α [ E ] for limit λ α<λ � L [ E ] = J α [ E ] α ∈ Ord Note that rud E denotes the closure under functions which are rudimentary in E (i.e. basic set operations like minus, union and pairing or intersection with E ). Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 10 / 29

Basic properties of L Condensation Let α be an infinite ordinal and let M ≺ ( L α , ∈ ) . Then the transitive collapse of M is equal to L β for some ordinal β ≤ α . Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 11 / 29

Basic properties of L Condensation Let α be an infinite ordinal and let M ≺ ( L α , ∈ ) . Then the transitive collapse of M is equal to L β for some ordinal β ≤ α . Comparison Let L α and L β for ordinals α and β be initial segments of L . Then one is an initial segment of the other, that means L α � L β or L β � L α . Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 11 / 29

Basic Concepts of Inner Model Theory Definition Let M be a countable model of set theory, κ a cardinal and U a κ -complete, nonprincipal ultrafilter on M . Then there is a transitive model N = Ult( M , U ) and an elementary embedding i U : M → N with critical point κ . We call N the ultrapower of M via U . Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 12 / 29

Basic Concepts of Inner Model Theory Definition Let M be a countable model of set theory, κ a cardinal and U a κ -complete, nonprincipal ultrafilter on M . Then there is a transitive model N = Ult( M , U ) and an elementary embedding i U : M → N with critical point κ . We call N the ultrapower of M via U . M Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 12 / 29

Basic Concepts of Inner Model Theory Definition Let M be a countable model of set theory, κ a cardinal and U a κ -complete, nonprincipal ultrafilter on M . Then there is a transitive model N = Ult( M , U ) and an elementary embedding i U : M → N with critical point κ . We call N the ultrapower of M via U . κ M Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 12 / 29

Basic Concepts of Inner Model Theory Definition Let M be a countable model of set theory, κ a cardinal and U a κ -complete, nonprincipal ultrafilter on M . Then there is a transitive model N = Ult( M , U ) and an elementary embedding i U : M → N with critical point κ . We call N the ultrapower of M via U . κ M N = Ult( M , U ) Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 12 / 29

Basic Concepts of Inner Model Theory Definition Let M be a countable model of set theory, κ a cardinal and U a κ -complete, nonprincipal ultrafilter on M . Then there is a transitive model N = Ult( M , U ) and an elementary embedding i U : M → N with critical point κ . We call N the ultrapower of M via U . i U ( κ ) i U κ M N = Ult( M , U ) Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 12 / 29

Basic Concepts of Inner Model Theory Mitchell and Jensen generalized the concept of measures to extenders to obtain stronger ultrapowers. Definition Let M be a countable model of set theory. An extender over M is a system of ultrafilters whose ultrapowers form a directed system, such that they give rise to a single elementary embedding. In fact for every embedding j : M → N there is an extender E over M which gives rise to this embedding. Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 13 / 29

Comparison One key concept of inner model theory is building iterated ultrapowers to compare two models. Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison One key concept of inner model theory is building iterated ultrapowers to compare two models. M Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison One key concept of inner model theory is building iterated ultrapowers to compare two models. E M Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison One key concept of inner model theory is building iterated ultrapowers to compare two models. E M N Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison One key concept of inner model theory is building iterated ultrapowers to compare two models. E M N Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison One key concept of inner model theory is building iterated ultrapowers to compare two models. E M N Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison One key concept of inner model theory is building iterated ultrapowers to compare two models. E M N Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison One key concept of inner model theory is building iterated ultrapowers to compare two models. E M N Ult( M , E ) Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison One key concept of inner model theory is building iterated ultrapowers to compare two models. E M N Ult( M , E ) Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison One key concept of inner model theory is building iterated ultrapowers to compare two models. E M N Ult( M , E ) Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Comparison One key concept of inner model theory is building iterated ultrapowers to compare two models. ⊲ E M N Ult( M , E ) Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 14 / 29

Iteration trees If the models contain Woodin cardinals, a linear iteration is not enough. Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 15 / 29

Iteration trees If the models contain Woodin cardinals, a linear iteration is not enough. We construct a tree of models with embeddings along the branches by building ultrapowers at successor steps and direct limits along the branches at limit steps. Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 15 / 29

Iteration trees If the models contain Woodin cardinals, a linear iteration is not enough. We construct a tree of models with embeddings along the branches by building ultrapowers at successor steps and direct limits along the branches at limit steps. M 0 Sandra Uhlenbrock Inner Models and Determinacy June 13th, 2016 15 / 29