Preliminaries Definition of least branch hod pair Comparison of least branch hod Least branch hod pairs pairs Hod pair capturing and HOD . John R. Steel University of California, Berkeley January 2017
Preliminaries Problem: Analyze HOD in models of determinacy. Definition of least branch hod pair Comparison of least branch hod Conjecture 1. Assume AD + + V = L ( P ( R )) ; then pairs Hod pair HOD | = GCH. capturing and HOD . = AD + + V = L ( P ( R )) such Conjecture 2. There is M | that HOD M | = “there is a subcompact cardinal”.
Preliminaries Problem: Analyze HOD in models of determinacy. Definition of least branch hod pair Comparison of least branch hod Conjecture 1. Assume AD + + V = L ( P ( R )) ; then pairs Hod pair HOD | = GCH. capturing and HOD . = AD + + V = L ( P ( R )) such Conjecture 2. There is M | that HOD M | = “there is a subcompact cardinal”. Definition “No long extenders” (NLE) is the assertion: there is no countable, iterable pure extender mouse with a long extender on its sequence.
Theorem Preliminaries Definition of least Suppose that κ is supercompact, and there are arbitrarily branch hod pair large Woodin cardinals. Suppose that V is uniquely Comparison of least branch hod iterable above κ ; then pairs (1) for any Γ ⊆ Hom ∞ such that L (Γ , R ) | = NLE , Hod pair capturing and HOD L (Γ , R ) | = GCH , and HOD . (2) there is a Γ ⊆ Hom ∞ such that HOD L (Γ , R ) | = “there is a subcompact cardinal”.
Theorem Preliminaries Definition of least Suppose that κ is supercompact, and there are arbitrarily branch hod pair large Woodin cardinals. Suppose that V is uniquely Comparison of least branch hod iterable above κ ; then pairs (1) for any Γ ⊆ Hom ∞ such that L (Γ , R ) | = NLE , Hod pair capturing and HOD L (Γ , R ) | = GCH , and HOD . (2) there is a Γ ⊆ Hom ∞ such that HOD L (Γ , R ) | = “there is a subcompact cardinal”. Moral: Below long extenders, there is a simple general notion of hod pair , and a general comparison theorem for them. They have a fine structure. Modulo the existence of iteration strategies , they can be used to analyze HOD , and they can have subcompact cardinals.
A Glossary Preliminaries Definition of least branch hod pair Comparison of least branch hod pairs (a) An extender E over M is a system of measures on M Hod pair capturing and coding an elementary i E : M → Ult ( M , E ) . E is short HOD . iff all its component measures concentrate on crit ( i E ) . Ult ( M , E ) = { [ a , f ] M E | f ∈ M and a ∈ [ λ ] <ω } , where λ = λ ( E ) = i E ( crit ( E )) .
Preliminaries Definition of least branch hod pair Comparison of least branch hod pairs Hod pair capturing and HOD .
Preliminaries Definition of least branch hod pair Comparison of least branch hod pairs (b) A normal iteration tree on M is an iteration tree T on Hod pair M in which the extenders used have increasing capturing and HOD . strengths, and are applied to the longest possible initial segment of the earliest possible model. (So along branches of T , generators are not moved.)
Preliminaries Definition of least branch hod pair Comparison of least branch hod pairs Hod pair capturing and HOD .
Preliminaries Definition of least branch hod pair Comparison of least branch hod pairs Hod pair capturing and HOD .
(c) An M-stack is a sequence s = �T 0 , ..., T n � of normal trees such that T 0 is on M , and T i + 1 is on the last model of T i . Preliminaries Definition of least branch hod pair Comparison of least branch hod pairs Hod pair capturing and HOD .
Preliminaries Definition of least branch hod pair (d) An iteration strategy Σ for M is a function that is Comparison of defined on M -stacks s that are by Σ whose last tree least branch hod pairs has limit length, and picks a cofinal wellfounded Hod pair capturing and branch of that tree. HOD .
Preliminaries Definition of least branch hod pair (d) An iteration strategy Σ for M is a function that is Comparison of defined on M -stacks s that are by Σ whose last tree least branch hod pairs has limit length, and picks a cofinal wellfounded Hod pair capturing and branch of that tree. HOD . (e) If s is an M -stack, then Σ s is the tail strategy given by Σ s ( t ) = Σ( s ⌢ t ) .
Preliminaries Definition of least branch hod pair (d) An iteration strategy Σ for M is a function that is Comparison of defined on M -stacks s that are by Σ whose last tree least branch hod pairs has limit length, and picks a cofinal wellfounded Hod pair capturing and branch of that tree. HOD . (e) If s is an M -stack, then Σ s is the tail strategy given by Σ s ( t ) = Σ( s ⌢ t ) . (f) It π : M → N is elementary, and Σ is an iteration strategy for N , then Σ π is the pullback strategy given by: Σ π ( s ) = Σ( π s ) .
Preliminaries Definition of least branch hod pair Comparison of least branch hod pairs Hod pair capturing and HOD .
Least branch hod pairs Definition Preliminaries A least branch premouse (lpm) is a structure M Definition of least M of extenders, branch hod pair constructed from a coherent sequence ˙ E Comparison of M for an iteration strategy for M . and a predicate ˙ least branch hod Σ pairs Hod pair capturing and HOD .
Least branch hod pairs Definition Preliminaries A least branch premouse (lpm) is a structure M Definition of least M of extenders, branch hod pair constructed from a coherent sequence ˙ E Comparison of M for an iteration strategy for M . and a predicate ˙ least branch hod Σ pairs Hod pair Remarks capturing and HOD . (a) M has a hierarchy, and a fine structure. By convention, there is a k = k ( M ) such that M is k-sound . (I.e., M = Hull k ( ρ M ∪ p M k ) .) k
Least branch hod pairs Definition Preliminaries A least branch premouse (lpm) is a structure M Definition of least M of extenders, branch hod pair constructed from a coherent sequence ˙ E Comparison of M for an iteration strategy for M . and a predicate ˙ least branch hod Σ pairs Hod pair Remarks capturing and HOD . (a) M has a hierarchy, and a fine structure. By convention, there is a k = k ( M ) such that M is k-sound . (I.e., M = Hull k ( ρ M ∪ p M k ) .) k M . (b) We use Jensen indexing for the extenders in ˙ E
Least branch hod pairs Definition Preliminaries A least branch premouse (lpm) is a structure M Definition of least M of extenders, branch hod pair constructed from a coherent sequence ˙ E Comparison of M for an iteration strategy for M . and a predicate ˙ least branch hod Σ pairs Hod pair Remarks capturing and HOD . (a) M has a hierarchy, and a fine structure. By convention, there is a k = k ( M ) such that M is k-sound . (I.e., M = Hull k ( ρ M ∪ p M k ) .) k M . (b) We use Jensen indexing for the extenders in ˙ E (c) At strategy-active stages α , we consider the M| α -least � ν, k , T � such that T is a normal tree of M| α , and limit length on M|� ν, k � that is by ˙ Σ M| α ( T ) is undefined. Then ˙ Σ M| ( α + 1 ) = ˙ M| α ∪ {� ν, k , T , b �} , where b is some ˙ Σ Σ cofinal branch of T .
Preliminaries Definition Definition of least A least branch hod pair (lbr hod pair) with with scope Z is branch hod pair a pair ( P , Σ) such that Comparison of least branch hod pairs (1) P is an lpm, Hod pair (2) Σ is an iteration strategy defined on all P -stacks capturing and HOD . s ∈ Z ,
Preliminaries Definition Definition of least A least branch hod pair (lbr hod pair) with with scope Z is branch hod pair a pair ( P , Σ) such that Comparison of least branch hod pairs (1) P is an lpm, Hod pair (2) Σ is an iteration strategy defined on all P -stacks capturing and HOD . s ∈ Z , Q ⊆ Σ s , and (3) if Q is a Σ -iterate of P via s , then ˙ Σ
Preliminaries Definition Definition of least A least branch hod pair (lbr hod pair) with with scope Z is branch hod pair a pair ( P , Σ) such that Comparison of least branch hod pairs (1) P is an lpm, Hod pair (2) Σ is an iteration strategy defined on all P -stacks capturing and HOD . s ∈ Z , Q ⊆ Σ s , and (3) if Q is a Σ -iterate of P via s , then ˙ Σ (4) Σ is self-consistent, normalizes well, and has strong hull condensation.
Preliminaries Definition Definition of least A least branch hod pair (lbr hod pair) with with scope Z is branch hod pair a pair ( P , Σ) such that Comparison of least branch hod pairs (1) P is an lpm, Hod pair (2) Σ is an iteration strategy defined on all P -stacks capturing and HOD . s ∈ Z , Q ⊆ Σ s , and (3) if Q is a Σ -iterate of P via s , then ˙ Σ (4) Σ is self-consistent, normalizes well, and has strong hull condensation. Σ is self-consistent iff the part of Σ that is a strategy for M|� ν, k � is consistent with the part of Σ that is a strategy for M|� µ, j � .
Normalizing well For �T , U� a stack on P , and W = W ( T , U ) its embedding Preliminaries normalization, we have Definition of least branch hod pair i T i U Comparison of P Q R least branch hod pairs Hod pair π capturing and i W HOD . S Then Σ 2-normalizes well iff �T , U� is by Σ iff W ( T , U ) is by Σ , and Σ π �W� = Σ �T , U� . for all such stacks �T , U� . Σ normalizes well iff all its tails 2-normalize well.
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