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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 ...
Binary198's user avatar
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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 ...
Rupert's user avatar
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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) \\ \...
Martín S's user avatar
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3 votes
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Do all limit $\alpha \in \omega_1^L$ satisfy $L_\alpha \models V=HC$?

In Gaps in the constructible universe, Marek and Srebrny, 1973 a gap ordinal and the start of a gap are defined as follows $\alpha$ is a gap ordinal iff $(L_{\alpha+1}-L_\alpha)\bigcap \mathcal{P}(\...
Martín S's user avatar
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Why can't $L_\beta$ contain a real coding a well-ordering of order-type $\beta$, when $\beta$ is a gap ordinal?

In Gaps in the constructible universe, Marek and Srebrny, 1973 a gap ordinal is defined as follows $\alpha$ is a gap ordinal iff $(L_{\alpha+1}-L_\alpha)\cap \mathcal{P}(\omega) = \emptyset$ Their ...
Martín S's user avatar
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11 votes
1 answer
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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 $V=L[r]$. ...
Asaf Karagila's user avatar
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3 votes
2 answers
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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 ...
Zoorado's user avatar
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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?
Someone211's user avatar
14 votes
1 answer
815 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 ...
Asaf Karagila's user avatar
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8 votes
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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) ...
Hanul Jeon's user avatar
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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 ...
Asaf Karagila's user avatar
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22 votes
2 answers
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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 $\mathbb{R}$...
Noah Schweber's user avatar
3 votes
4 answers
527 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 ...
Zhiwei's user avatar
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6 votes
1 answer
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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 ...
Asaf Karagila's user avatar
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3 votes
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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 ...
Ralf Schindler's user avatar
9 votes
1 answer
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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 ...
James E Hanson's user avatar
6 votes
1 answer
542 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) ...
Monroe Eskew's user avatar
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4 votes
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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 ...
Julian Barathieu's user avatar
19 votes
1 answer
466 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 ...
Alexander Block's user avatar
10 votes
4 answers
548 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 ...
Noah Schweber's user avatar
4 votes
2 answers
276 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 ...
Christopher Menzel's user avatar
3 votes
2 answers
517 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 ...
Maxime Ramzi's user avatar
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7 votes
1 answer
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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 $...
Alon Navon's user avatar
10 votes
0 answers
358 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 ...
Mohammad Golshani's user avatar
10 votes
1 answer
572 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 ...
Mohammad Golshani's user avatar
10 votes
1 answer
773 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 ...
Noah Schweber's user avatar
7 votes
0 answers
246 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 ...
Mohammad Golshani's user avatar
2 votes
2 answers
241 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 ...
tww's user avatar
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17 votes
2 answers
801 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), ...
Noah Schweber's user avatar
7 votes
1 answer
340 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 ...
Cody Dance's user avatar
4 votes
1 answer
188 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 $T$ ...
Jesse Elliott's user avatar
11 votes
1 answer
1k 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 ...
David Roberts's user avatar
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9 votes
1 answer
710 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_{\...
Mohammad Golshani's user avatar
7 votes
1 answer
354 views

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 $\varphi$ ...
Noah Schweber's user avatar
3 votes
1 answer
289 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 ...
user avatar
7 votes
2 answers
360 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 ‎...
user avatar
4 votes
1 answer
344 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 ...
Noah Schweber's user avatar