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Results tagged with inner-model-theory
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user 102684
The study of canonical inner models for large cardinal hypotheses, with particular attention to their fine structure theory, and iterability issues.
6
votes
What happens with large singular cardinals on the far side of the HOD dichotomy?
This is independent relative to the failure of the HOD hypothesis in the presence of large cardinals.
We first give a positive answer under GCH. (Note that if it is consistent for the HOD Hypothesis …
- 8,099
6
votes
Existence of inner models of $\mathrm{ZFC} \ +$ forcing axioms, under incompatible assumptions
I want to comment that both AD and "$V$ is $L$-like" are consistent with the existence of inner models of very strong theories.
For AD, this is actually quite simple. Suppose there is a supercompact c …
- 8,099
5
votes
Accepted
Do precipitous ideals "always" come from collapsing?
If $M_1^\#$ exists and is fully iterable, then there is an inner model $M$ in which $\omega_1$ is measurable and an $M$-generic $G$ such that $V_{\omega_1+2}\in M[G].$ Just iterate the first normal ul …
- 8,099
9
votes
Accepted
Uniqueness of countable version of $L[U]$?
Andrés's comment can be turned into an answer, but there is a slight subtlety. Suppose $M$ is a countable iterable model of ZFC + $V = L[U]$. Let $\kappa$ be its measurable cardinal and $\alpha$ be it …
- 8,099
8
votes
Why "adding" a single extender cannot give an L-like inner model for say, a strong cardinal?
A fairly complete answer to this question appears in Woodin's "In search of Ultimate $L$" at the beginning of the section on Martin-Steel extender sequences. Woodin omits the proofs, so I'll fill in s …
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5
votes
Accepted
Consistency strength of lifting through a lot of collapsing
$\text{AD}^{L(\mathbb R)}$ suffices. The situation actually holds in the model $H = \text{HOD}^{L(\mathbb R)}$. We will have $\kappa = \omega_1$ and $j : H\to \text{Ult}(H,U)$ equal to the ultrapower …
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16
votes
Forcings predicted by core model theory $+$$ZFC$ results proved by the method of core model ...
Question 1: A good example is Woodin's extender algebra. One reference describing the discovery of the extender algebra is the introduction to Neeman's book The Determinacy of Long Games. I am basical …
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5
votes
Accepted
Pointwise definable models of determinacy
The least ordinal $\kappa$ such that $L_\kappa(\mathbb R)$ satisfies KP is a counterexample. This essentially reduces to Theorem 1.3 of Tony Martin's "The Largest Countable This, That, and the Other." …
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8
votes
Accepted
Inner model with a $\mathit{\Delta}^1_3$-good well-ordering of the reals
The situation is a bit more complicated than you might hope because of the periodicity phenomena in the projective hierarchy. For odd $n$, assuming $\mathbf{\Delta}^1_{n-1}$-determinacy, the set $Q_n$ …
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13
votes
A “paradox” about the inner model problem
The issue is Thesis 2. The notion of a canonical inner model is vague, but I don't think that when inner model theorists use the term they mean to suggest that the model is obtained by iterating a sha …
- 8,099
12
votes
Accepted
Does $Π^1_{2n+1} = (Σ^1_{2n+1})^{M_{2n-1}}$?
Your conjecture is true. We assume PD throughout. The proof I see requires citing a number of facts from inner model theory and descriptive set theory. First, it uses Woodin's theorem characterizing t …
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