# Questions tagged [inner-models]

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

23 questions
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### Arithmetic sums of Marczewski null sets

First, recall the Marczewski ideal, called $s^0$: a set $A$ of reals is in $s^0$ iff for every perfect set $P$ there is some perfect $P' \subset P$ such that $P' \cap A = \emptyset$. Secondly, by way ...
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### Is there a minimal extension of $L$ that is not a forcing extension?

It's well known that Sacks forcing constructs a real of minimal constructability degree, i.e. a real $x$ such that for any $y\in L(x) \setminus L$, $L(y) = L(x)$. It's also well known that certain ...
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### fake and weak cardinals

Suppose $\lambda$ is a successor of a singular cardinal. We will say $\lambda$ fake if there is a transitive set $M$ such that $\lambda \subseteq M$ satisfying $\mathrm{ZFC}^-$ (ZFC without powerset) ...
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### What does $L(A,\mathbb{R})$ mean?

I many papers by Woodin, and on some answers here on MathOverflow (like the first answer of this question), I see the expression "$L(A,\mathbb{R})$" being used, but I have never seen it defined. I ...
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### Is every transitive ZF-model of inaccessible height a truncation of an inner model?

Let $\kappa$ be an inaccessible cardinal and let $M \subseteq V_{\kappa}$ be an inner model of $V_{\kappa}$, i.e., a transitive model of $\mathsf{ZF}$ containing all the ordinals up to $\kappa$. My ...
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### What are some kinds of models where DC holds?

There are a lot of ways to build a model where DC fails. However, all of them that I'm aware of involve adding at least a messy set of reals (or rather, taking a forcing extension and then passing to ...
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### Independence of a certain set w.r.t. Z set theory with urelements

Skolem proved the need for the axiom schema of Replacement by, in essence, showing that $V_{ω+ω}$ is a model of Zermelo's set theory Z (the smallest "natural" model) and, hence, that the existence of ...
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### Relation between AC and the axiom of foundation

The fact that the axiom of foundation doesn't imply the axiom of choice is pretty standard (the model Cohen created to prove the consistency of $\neg AC$ models the axiom of foundation as well), and ...
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### Is the consistency of $\mathcal{L}_{\infty\omega}$-sentences absolute?

The question is exactly that of the title. Suppose $\varphi\in V$ is an $\mathcal{L}_{\infty\omega}$-sentence, and $W$ is an inner model of $V$ such that $\varphi\in W$. Is the statement $\varphi$ ...
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### Do inner models of unique measurable cardinals have a regular behavior? (Edited and Revised Version)

We know that if ‎$‎‎\kappa‎‎$ is a‎ ‎measurable ‎cardinal ‎and ‎‎$‎‎\mu$ be a‎ ‎two-valued ‎non-trivial‎‎$‎‎‎\kappa‎$-additive ‎measure ‎on ‎it ‎the‎n the corresponding inner model produced by ...
Definition (1): A‎n‎‎ ‎inner ‎model ‎of ‎‎$‎ZFC‎$ ‎is a‎ ‎tarnsitive proper class ‎model ‎‎of $‎‎ZFC$ ‎which ‎contains ‎all ‎ordinal numbers. ‎Informally ‎we ‎denote ‎the ‎collection ‎of ‎all ‎inner ‎...
Consider two ultrafilters, $U$ and $V$, on the same cardinal $\kappa$. Let $D(U, V)=\lbrace X\subseteq \kappa: X\in U-V\rbrace$; clearly $D(U, V)$ is a lattice under $\subseteq, \cap, \cup$ since the ...