# Questions tagged [inner-models]

An inner model is a transitive proper class substructure of the universe of sets, that satisfies $\mathsf{ZF}(\mathsf{C})$.

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### Why do we need the comparison lemma?

An inner model is a standard transitive (proper class) structure which satisfies all the axioms of ZFC and contains all the ordinals. The simplest and most well-known inner model is Gödel’s $L$, which ...
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### Question about notation involving HOD

This is a rather naive question about notation. From the sources I have been able to find I have difficulty working out the difference between $HOD_{x}$ and $HOD[x]$ (because I cannot actually find ...
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### Why are the sharps of sets of big ordinals not in $\mathcal{P}(\omega)$?

In his talk A Condensed History of Condensation, Welch presents the following recursive sharp function, that is total when all sharps exist: \begin{align*} \# \colon ON &\to \mathcal{P}(ON) \\ \...
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### Inner models and strongly compact cardinals

The following question is motivated by a result of Magidor that it is consistent that the least strongly compact cardinal is the least measurable cardinal. Question. Assume $\kappa$ is a strongly ...
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### Singular in $V$ regular in $HOD$

Prikry forcing can be used to produce a model $V$ of $ZFC$ such that fo rsome cardinal $\kappa$ we have: (1) $\kappa$ is singular in $V$ of cofinality $\omega,$ (2) $\kappa$ is regular (and in fact ...
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### Can an ultrapower be undone by forcing?

I am not 100% certain this question is appropriate for MO; I may just be missing something obvious. Also, I vaguely recall a similar question being asked here a while ago, but I can't find it; if it ...
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### $V$ as a $HOD$ of its class generic extension

By an old result of Roguski, The theory of the class $HOD$, any model $V$ of $ZFC$ has a class generic extension $V[G]$ such that $HOD$ of $V[G]$ equals $V$. This result is also stated and generalized ...
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### Precise interpretability strength of $\mathcal P_{DF}(\omega)$ and $L_{\omega_1^{CK}}$

I am curious about the relationship between the definable power set of $\omega$ and the $\omega_1^{CK}$th level of the constructible sets $L$. In short, $\omega_1^{CK}$ is the least nonrecursive ...
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### Whatever happened to $L(j)$?

So this question probably shows my inner model theoretic ignorance, but: In "Two remarks on elementary embeddings of the universe" (http://projecteuclid.org/download/pdf_1/euclid.pjm/1102969567), ...
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### Characterizing L(R) Cardinals in HOD

We're working in L(R) under AD. We know that $\omega_1$ is the least measurable in HOD, $\Theta$ is the least woodin, $\delta^2_1$ is the least strong to the woodin, etc. My question is about ...
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### Classify set theories whose transitive models sharing the same sets of ordinals are equal

This question is a follow-up from my recent question, Classifying set theories whose standard models sharing the same ordinals are equal Let's say that a (recursively axiomatizable) set theory $T$ ...
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### Existence of internal toposes/inner models in a topos

It has been known for some time that one can define a topos as a model of a (finitary) essentially algebraic theory (or in other words, can be defined internal to any category with finite limits). In ...
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