# 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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questions

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### Coding the universe into a real over better core models

One of the most incredible results in modern set theory, due to Jensen, is that given any model of $\sf ZFC$, there is a class forcing which adds a real number $r$ and in the extension $V=L[r]$. ...

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### Existence of inner models of $\mathrm{ZFC} \ +$ forcing axioms, under incompatible assumptions

I am curious about the existence of inner models of $\mathrm{ZFC}$ in conjunction with forcing axioms, under assumptions inconsistent with such theories. For example:
can we prove under any extension ...

**8**

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228 views

### Inner model theory without choice

How much of the inner model project can be constructed without assuming the axiom of choice? I.e. which large cardinals provably have canonical inner models not assuming choice?

**14**

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676 views

### Is there a minimal inner model for determinacy?

Assume $\sf ZF+AD$. Is there some inner model $M$ containing all the ordinals such that $M\models\sf ZF+AD$ as well?
What if we require $\omega_1$ and/or $\omega_2$ to be computed correctly?
Can we ...

**7**

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422 views

### Is there an abstract logic that defines the mantle?

It is a known result by Scott and Myhill that the second-order version of $L$ yields $\mathrm{HOD}$.
Recently, Kennedy, Magidor, and Väänänen (Inner models from extended logics: Part I and II) ...

**10**

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247 views

### How wealthy are canonical inner models?

One of the way a person shows their wealth is by having many diamonds. The same can be said about models of $\sf ZFC$. We can add generic diamond sequences, while preserving the old ones, so in some ...

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492 views

### Gently changing measure

This question was asked and bountied on MSE without answer, so I'm porting it here:
There's an easy way to change the measure of a set of reals by moving to a larger universe: simply make $\mathbb{R}$...

**3**

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**4**answers

479 views

### Is a sentence true for two substructures also true for their intersection? [closed]

Let $L$ be a first order language and $M$ an interpretation of $L$.
If $A$ and $B$ are two substructures of $M$ and their intersection $C=A\cap B\ne \emptyset$, then is it the case that every sentence ...

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212 views

### Generic saturation of inner models

Say that an inner model $M$ of $V$ is generically saturated if for every forcing notion $\Bbb P\in M$, either there is an $M$-generic for $\Bbb P$ in $V$, or forcing with $\Bbb P$ over $V$ collapses ...

**3**

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200 views

### 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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181 views

### 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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483 views

### 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) ...

**4**

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240 views

### 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 ...

**19**

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434 views

### 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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522 views

### 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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262 views

### 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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418 views

### 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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241 views

### Extending Sacks forcing

Sacks forcing allows us to build a model $V[G]$, such that there is no "intermediate model" between $V$ and $V[G]$, meaning if $V \subseteq W \subseteq V[G]$ is a model of ZFC then either $W = V$ or $...

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313 views

### 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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515 views

### 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 ...

**8**

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564 views

### 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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221 views

### $V$ 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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212 views

### 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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546 views

### 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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305 views

### 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 ...

**4**

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168 views

### 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 $T$ ...

**10**

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**1**answer

905 views

### 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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614 views

### Reinhardt cardinals and iterability

Work in $ZF$. Let $j:V\to V$ be a non-trivial elementary embedding which is iterable, so that we can iterate it and form models $M_\alpha, \alpha\in ON,$ with $M_0=V,$ and elementary embeddings $j_{\...

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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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254 views

### 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 ...

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306 views

### Is there an inner model between two distinct inner models of ZFC?

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 ...

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313 views

### Lattice of differences between ultrafilters

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 ...