The interplay between inner model theory and descriptive set theory in a nutshell Sandra M¨ uller Universit¨ at Wien June 2019 Logic Fest in the Windy City Sandra M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 1
D I n e n t C e e r a r o C m n M n a o i n n r o D d a i d c i e c n e s a y a l c l s l r a s I w i n p n f n i t d r t i e o h v i r m e t L s M S a D I r e m o g t e d p t e e T e a l r s h c m t e o i n r a y c y Sandra M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 2
Games in Set Theory Definition (Gale-Stewart 1953) Let A ⊂ ω ω . We denote the following game by G ( A ) I n 0 n 2 . . . for n k ∈ ω for all k ∈ ω . II n 1 n 3 . . . We say player I wins the game iff ( n k ) k ∈ ω ∈ 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 M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 3
Which games are determined and what is it good for? Theorem (Gale-Stewart, 1953) ( AC ) Let A ⊂ ω ω be open or closed. Then G ( A ) is determined. Sandra M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 4
Which games are determined and what is it good for? Theorem (Gale-Stewart, 1953) ( AC ) Let A ⊂ ω ω 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 M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 4
Which games are determined and what is it good for? Theorem (Gale-Stewart, 1953) ( AC ) Let A ⊂ ω ω 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. Determinacy implies regularity properties. Theorem (Mycielski, Swierczkowski, Mazur, Davis) If all sets of reals are determined, then all sets of reals are Lebesgue measurable, have the Baire property, and have the perfect set property. Sandra M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 4
Determinacy for Definable Sets of Reals Theorem (Martin, 1975) Assume ZFC . Then every Borel set of reals is determined. Sandra M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 5
Determinacy for Definable Sets of Reals Theorem (Martin, 1975) Assume ZFC . Then every Borel set of reals is determined. We define the projective hierarchy from Borel sets 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 M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 5
Determinacy for Definable Sets of Reals Theorem (Martin, 1975) Assume ZFC . Then every Borel set of reals is determined. Determinacy for all analytic sets of reals is not provable in ZFC alone. Theorem (Martin, 1970) Assume ZFC and that there is a measurable cardinal. Then every analytic set is determined. Sandra M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 5
Determinacy for Definable Sets of Reals Theorem (Martin, 1975) Assume ZFC . Then every Borel set of reals is determined. Determinacy for all analytic sets of reals is not provable in ZFC alone. Theorem (Martin, 1970) Assume ZFC and that there is a measurable cardinal. Then every analytic set is determined. 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 M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 5
Determinacy for Definable Sets of Reals Theorem (Martin, 1975) Assume ZFC . Then every Borel set of reals is determined. Determinacy for all analytic sets of reals is not provable in ZFC alone. Theorem (Martin, 1970) Assume ZFC and that there is a measurable cardinal. Then every analytic set is determined. 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. Are large cardinals necessary for the determinacy of these sets of reals? How can these large cardinals affect what happens with the sets of reals? Sandra M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 5
D I n e n t C e e r a r o C m n M n a o i n n r o D d a i d c i e c n e s a y a l c l s l r a s I w i n p n f n i t d r t i e o h v i r m e t L s M S a D I r e m o g t e d p t e e T e a l r s h c m t e o i n r a y c y Sandra M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 6
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 M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 7
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 M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 8
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 M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 9
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 M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 9
A First Equivalence for Analytic Determinacy Definition Let x be a real. We say x # exists iff for some limit ordinal λ , the model L λ [ x ] has an uncountable set Γ of indiscernibles, i.e. for n < ω and any two increasing sequences ( α 0 , . . . , α n ) and ( β 0 , . . . , β n ) from Γ and any formula ϕ , L λ [ x ] � ϕ ( x, α 0 , . . . , α n ) ⇔ L λ [ x ] � ϕ ( x, β 0 , . . . , β n ) . Sandra M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 10
A First Equivalence for Analytic Determinacy Definition Let x be a real. We say x # exists iff for some limit ordinal λ , the model L λ [ x ] has an uncountable set Γ of indiscernibles, i.e. for n < ω and any two increasing sequences ( α 0 , . . . , α n ) and ( β 0 , . . . , β n ) from Γ and any formula ϕ , L λ [ x ] � ϕ ( x, α 0 , . . . , α n ) ⇔ L λ [ x ] � ϕ ( x, β 0 , . . . , β n ) . Theorem (Harrington, Martin) The following are equivalent. (a) All analytic games are determined. (b) x # exists for all reals x . Sandra M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 10
A First Equivalence for Analytic Determinacy Definition Let x be a real. We say x # exists iff for some limit ordinal λ , the model L λ [ x ] has an uncountable set Γ of indiscernibles, i.e. for n < ω and any two increasing sequences ( α 0 , . . . , α n ) and ( β 0 , . . . , β n ) from Γ and any formula ϕ , L λ [ x ] � ϕ ( x, α 0 , . . . , α n ) ⇔ L λ [ x ] � ϕ ( x, β 0 , . . . , β n ) . Theorem (Harrington, Martin) The following are equivalent. (a) All analytic games are determined. (b) x # exists for all reals x . To see how this relates to measurable cardinals, we need to look at a different definition of x # . Sandra M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 10
Basic Concepts of Inner Model Theory Definition Let M be a transitive model of set theory, κ a cardinal in M 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 M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 11
Basic Concepts of Inner Model Theory Definition Let M be a transitive model of set theory, κ a cardinal in M 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 M¨ uller (Universit¨ at Wien) Inner model theory and determinacy June 2019 11
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