M 1 and its strategy extensions (and beyond) Farmer Schlutzenberg, - PowerPoint PPT Presentation

M 1 and its strategy extensions (and beyond) Farmer Schlutzenberg, University of Münster 15th International Luminy Workshop in Set Theory September 26, 2019 M 1 and its strategy extensions

We will discuss connections between (pure extender) mice, and strategy mice. Some key things: 1. How much of its own iteration strategy can be added to M 1 without destroying the Woodinness of δ ? M 1 and its strategy extensions

We will discuss connections between (pure extender) mice, and strategy mice. Some key things: 1. How much of its own iteration strategy can be added to M 1 without destroying the Woodinness of δ ? 2. Given an M 1 -cardinal κ > δ , what is the κ -mantle of M 1 ? M 1 and its strategy extensions

Background... Definition 1.1. (Pre-)mice M : M 1 and its strategy extensions

Background... Definition 1.1. (Pre-)mice M : – M = L α [ E ] , M 1 and its strategy extensions

Background... Definition 1.1. (Pre-)mice M : – M = L α [ E ] , � � E M – E = α<λ is a good sequence of extenders, α M 1 and its strategy extensions

Background... Definition 1.1. (Pre-)mice M : – M = L α [ E ] , � � E M – E = α<λ is a good sequence of extenders, α – M is iterable (mouse). M 1 and its strategy extensions

Background... Definition 1.1. (Pre-)mice M : – M = L α [ E ] , � � E M – E = α<λ is a good sequence of extenders, α – M is iterable (mouse). With extender E = E M α form (internal) ultrapower U = Ult 0 ( M , E ) , M 1 and its strategy extensions

Background... Definition 1.1. (Pre-)mice M : – M = L α [ E ] , � � E M – E = α<λ is a good sequence of extenders, α – M is iterable (mouse). With extender E = E M α form (internal) ultrapower U = Ult 0 ( M , E ) , gives Σ 1 -elementary ultrapower map i M E : M → U M 1 and its strategy extensions

Background... Definition 1.1. (Pre-)mice M : – M = L α [ E ] , � � E M – E = α<λ is a good sequence of extenders, α – M is iterable (mouse). With extender E = E M α form (internal) ultrapower U = Ult 0 ( M , E ) , gives Σ 1 -elementary ultrapower map i M E : M → U (ignoring details). M 1 and its strategy extensions

Background... Definition 1.1. (Pre-)mice M : – M = L α [ E ] , � � E M – E = α<λ is a good sequence of extenders, α – M is iterable (mouse). With extender E = E M α form (internal) ultrapower U = Ult 0 ( M , E ) , gives Σ 1 -elementary ultrapower map i M E : M → U (ignoring details). Set M 0 = M and M 1 = U ... M 1 and its strategy extensions

Iteration trees T on M : – Given M β , choose E β ∈ E M β , choose α ≤ β , set M β + 1 = Ult ( M α , E β ) , M 1 and its strategy extensions

Iteration trees T on M : – Given M β , choose E β ∈ E M β , choose α ≤ β , set M β + 1 = Ult ( M α , E β ) , – embedding i α,β + 1 : M α → M β + 1 , M 1 and its strategy extensions

Iteration trees T on M : – Given M β , choose E β ∈ E M β , choose α ≤ β , set M β + 1 = Ult ( M α , E β ) , – embedding i α,β + 1 : M α → M β + 1 , – α = tree-predecessor of β + 1. M 1 and its strategy extensions

Iteration trees T on M : – Given M β , choose E β ∈ E M β , choose α ≤ β , set M β + 1 = Ult ( M α , E β ) , – embedding i α,β + 1 : M α → M β + 1 , – α = tree-predecessor of β + 1. – Model M α at node α ∈ T .

Iteration trees T on M : – Given M β , choose E β ∈ E M β , choose α ≤ β , set M β + 1 = Ult ( M α , E β ) , – embedding i α,β + 1 : M α → M β + 1 , – α = tree-predecessor of β + 1. – Model M α at node α ∈ T . M 0

Iteration trees T on M : – Given M β , choose E β ∈ E M β , choose α ≤ β , set M β + 1 = Ult ( M α , E β ) , – embedding i α,β + 1 : M α → M β + 1 , – α = tree-predecessor of β + 1. – Model M α at node α ∈ T . M 0 ∋ E 0

Iteration trees T on M : – Given M β , choose E β ∈ E M β , choose α ≤ β , set M β + 1 = Ult ( M α , E β ) , – embedding i α,β + 1 : M α → M β + 1 , – α = tree-predecessor of β + 1. – Model M α at node α ∈ T . M 1 = Ult ( M 0 , E 0 ) i 01 M 0 ∋ E 0

Iteration trees T on M : – Given M β , choose E β ∈ E M β , choose α ≤ β , set M β + 1 = Ult ( M α , E β ) , – embedding i α,β + 1 : M α → M β + 1 , – α = tree-predecessor of β + 1. – Model M α at node α ∈ T . M 1 ∋ E 1 i 01 M 0 ∋ E 0

Iteration trees T on M : – Given M β , choose E β ∈ E M β , choose α ≤ β , set M β + 1 = Ult ( M α , E β ) , – embedding i α,β + 1 : M α → M β + 1 , – α = tree-predecessor of β + 1. – Model M α at node α ∈ T . M 2 i 12 M 1 ∋ E 1 i 01 M 0 ∋ E 0

Iteration trees T on M : – Given M β , choose E β ∈ E M β , choose α ≤ β , set M β + 1 = Ult ( M α , E β ) , – embedding i α,β + 1 : M α → M β + 1 , – α = tree-predecessor of β + 1. – Model M α at node α ∈ T . M 2 ∋ E 2 i 12 M 1 ∋ E 1 i 01 M 0 ∋ E 0 M 1 and its strategy extensions

Iteration trees T on M : – Given M β , choose E β ∈ E M β , choose α ≤ β , set M β + 1 = Ult ( M α , E β ) , – embedding i α,β + 1 : M α → M β + 1 , – α = tree-predecessor of β + 1. – Model M α at node α ∈ T . M 3 M 2 ∋ E 2 i 13 i 12 M 1 ∋ E 1 i 01 M 0 ∋ E 0 M 1 and its strategy extensions

Iteration trees T on M : – Given M β , choose E β ∈ E M β , choose α ≤ β , set M β + 1 = Ult ( M α , E β ) , – embedding i α,β + 1 : M α → M β + 1 , – α = tree-predecessor of β + 1. M 4 – Model M α at node α ∈ T . M 3 M 2 ∋ E 2 i 04 i 13 i 12 M 1 ∋ E 1 i 01 M 0 ∋ E 0 M 1 and its strategy extensions

Iteration trees T on M : – Given M β , choose E β ∈ E M β , choose α ≤ β , set M 5 M β + 1 = Ult ( M α , E β ) , – embedding i α,β + 1 : M α → M β + 1 , – α = tree-predecessor of β + 1. M 4 i 25 – Model M α at node α ∈ T . M 3 M 2 ∋ E 2 i 04 i 13 i 12 M 1 ∋ E 1 i 01 M 0 ∋ E 0 M 1 and its strategy extensions

Iteration trees T on M : – Given M β , choose E β ∈ E M β , choose α ≤ β , set M 5 M β + 1 = Ult ( M α , E β ) , – embedding i α,β + 1 : M α → M β + 1 , – α = tree-predecessor of β + 1. M 4 i 25 i 15 – Model M α at node α ∈ T . M 3 M 2 ∋ E 2 i 04 i 13 i 12 M 1 ∋ E 1 i 01 M 0 ∋ E 0 M 1 and its strategy extensions

Iteration trees T on M : – Given M β , choose E β ∈ E M β , choose α ≤ β , set M 5 M β + 1 = Ult ( M α , E β ) , – embedding i α,β + 1 : M α → M β + 1 , – α = tree-predecessor of β + 1. M 4 i 25 i 15 – Model M α at node α ∈ T . – Write M T α = M α , E T α = E α , etc. M 3 M 2 ∋ E 2 i 04 i 13 i 12 M 1 ∋ E 1 i 01 M 0 ∋ E 0 M 1 and its strategy extensions

Iteration trees T : – Limit stages λ ? M 1 and its strategy extensions

Iteration trees T : – Limit stages λ ? Choose cofinal branch b , set M λ = M b = direct limit of models along b . M 1 and its strategy extensions

Iteration trees T : – Limit stages λ ? Choose cofinal branch b , set M λ = M b = direct limit of models along b . – Iteration strategy Σ chooses branches b , guarantees wellfounded models. M 1 and its strategy extensions

Iteration trees T : – Limit stages λ ? Choose cofinal branch b , set M λ = M b = direct limit of models along b . – Iteration strategy Σ chooses branches b , guarantees wellfounded models. – M is (fully) iterable if such a Σ exists. M 1 and its strategy extensions

Definition 1.2. An iteration tree T is normal iff ⇒ lh ( E T α ) ≤ lh ( E T α < β = β ) , and all extenders apply to the earliest and largest model possible. We write lh ( T ) for the length of T . T is on M T 0 . If T has successor length then M T ∞ = last model of T . If b is a branch through T then M T b = direct limit model along b , and i T b : M T 0 → M T b is the direct limit map. Let T be a normal iteration tree, of limit length. Then (Coherence) ⇒ M T α || lh ( E T α ) = M T β || lh ( E T α < β = α ) . Write: – δ ( T ) = sup α< lh ( T ) lh ( E T α ) . – M ( T ) = eventual model of agreement of height δ ( T ) , � M T α || lh ( E T M ( T ) = α ) . α< lh ( T ) M 1 and its strategy extensions

Definition 1.3. M 1 = the minimal proper class mouse with a Woodin cardinal δ = δ M 1 . Then: – M 1 = L [ E M 1 ] = L [ E ] with E ⊆ M 1 | δ . – Write Σ M 1 for the (unique) normal iteration strategy for M 1 . – Is M 1 closed under Σ M 1 ? – If T is a normal tree on M 1 , we say T is maximal iff L [ M ( T )] | = “ δ ( T ) is Woodin” . – Let T be maximal and b = Σ M 1 ( T ) . Then M T b = L [ M ( T )] , b ( δ M 1 ) = δ ( T ) . i T – M 1 computes correct branches through non-maximal trees. M 1 and its strategy extensions

But there are maximal trees U ∈ M 1 such that: – L [ M ( U )] is a ground of M 1 , – δ ( U ) is a successor cardinal of M 1 , – so i U b / ∈ M 1 , – so b / ∈ M 1 (uses Woodin’s genericity iterations). (Corollary: M 1 has proper grounds.) M 1 and its strategy extensions

More generally, mice with Woodin cardinals cannot iterate themselves. M 1 and its strategy extensions

More generally, mice with Woodin cardinals cannot iterate themselves. We can build structures which can: Definition 1.4. A strategy mouse* is an iterable structure M = L α [ E , Σ] M 1 and its strategy extensions