# Questions tagged [inner-model-theory]

The study of canonical inner models for large cardinal hypotheses, with particular attention to their fine structure theory, and iterability issues.

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### Consistency of reflective sequences

Is it consistent that there is a measurable cardinal $κ$, a $κ$-complete normal nonprincipal ultrafilter $U$ on $κ$, and $S∈U$ such that for every $T⊂S$ with $T∈U$ and $T$ ordinal definable from $S$ ...

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### Universally meager spaces and large cardinals

Definition: (Todorcevic) A subset $A$ of a topological space $X$ is called universally meager if for every Baire space $Y$ and every continuous $f : Y \to X$ which is nowhere constant (not constant on ...

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### Does $0^{\#}$ imply that $L$'s set generic multiverse has irreconcilable theories?

Suppose $0^{\#}$ exists. Let $M(L)$ be the generic multiverse over $L$ as viewed in $V$, i.e. $M(L)$ consists of all $L[g]$ where $g \in V$ is $\mathbb{P}$-generic over $L$ for some forcing $\mathbb{P}...

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### What is the modal logic of outer multiverse?

The mathematical multiverse could be viewed as a gigantic Kripke model with models of $ZFC$ as possible worlds connected to each other via a certain accessibility relation.
The modal logic associated ...

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### Does $0^{\#}$ imply the failure of upward directedness in the set generic universe over $L$?

Question 1. Suppose that $0^{\#}$ exists. Are there set generic filters $g,h$ over $L$ ($g,h \in V$) such that for any $f$ ($\in V$) which is generic over $L$ we have that $L[g] \cup L[h] \not \...

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### Forcings predicted by core model theory $+$$ZFC$ results proved by the method of core model theory

I have two unrelated question.
First question. To motivate the question, let me explain an example. The natural way to force the failure of singular cardinals hypothesis ($SCH$), is to start with a ...

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### Taking a proper class as a model for Set Theory

When I am reading through higher Set Theory books I am frequently met with statements such as '$V$ is a model of ZFC' or '$L$ is a model of ZFC' where $V$ is the Von Neumann Universe, and $L$ the ...

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### Is $V=\textsf{HOD}\not\Rightarrow\textsf{GCH}$ consistent?

Whenever $M$ is some fine-structural $L$-like model we can prove the implication $V=M\Rightarrow\textsf{GCH}$. For $L$ this is due to Gödel, and for the modern extender models it follows simply by ...

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### A Weak form of Extendibility and Inner Model Theory

Let a cardinal $\kappa$ be $n$-shadow iff $\kappa$ does not have cofinality $\omega$ and for any $n$-th order sentence $\varphi$ in the language $\mathcal{L}_\in$, $\varphi\Leftrightarrow V_\kappa\...

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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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### Reals which must, can't or might be added by forcing

Let $W \subseteq V$ be an inner model of ZFC. There are a variety of theorems that characterize when a real $x \in V$ is the generic of a forcing notion $\mathbb P \in W$, for example, the ...

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### Complexity of $L[\mathrm{cf}]$

Assuming large cardinal axioms, which real numbers are in $L[\mathrm{cf}]$, where $\mathrm{cf}$ is the cofinality function on ordinals?
$L[\mathrm{cf}]$ is the minimal inner model that 'knows' the ...

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### The Ultimate L in a Nutshell: On Descriptive Articles

Everybody who catches a fleeting glimpse of Woodin's central papers on Ultimate $L$ (i.e. Suitable Extender Models I & II), admits that they aren't so tempting for lazy readers who don't like to ...

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### Is there a natural inner model of AD$_\mathbb{R}$?

The question is as in the title, but let me explain a bit.
Assuming a proper class of Woodin cardinals, $L(\mathbb{R})$ satisfies AD (and DC). And $L(\mathbb{R})$ is a very natural inner model. I'm ...

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### How much real determinacy can live in $L(\mathbb{R})$?

It's well-known that AD$_\mathbb{R}$ fails in $L(\mathbb{R})$, provably in ZFC. This is because:
AD$^{L(\mathbb{R})}$ implies DC$^{L(\mathbb{R})}$.
Over ZF+DC, AD + "Every set of reals has a scale" ...

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### Definition of $M_n^\sharp(X)$ for arbitrary set $X$

The definition of $M_n^\sharp$ in [OIMT10] is the unique sound, $(\omega,\omega_1,\omega_1+1)$-iterable mouse which is not $n$-small, but all of whose proper initial segments are $n$-small.
What is ...

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### What is a 'power admissible model'?

Q: What exactly is a power admissible model?
Background: Admissible models, introduced by Jon Barwise, form the building blocks of inner model theory. They are transitive models $\mathcal M = (M; \in)...

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### Proper class of Woodins and $\textsf{AD}_{\mathbb R}$-hypothesis

The $\textsf{AD}_{\mathbb R}$-hypothesis is the statement that there is a $\lambda$ which is both a limit of Woodins and a limit of ${<}\lambda$-strongs. Are there any results relating the ...

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### Absoluteness for the Chang model

Often, large cardinals imply that many definable inner models and transitive sets have theories which are absolute under forcing. For example, assuming a proper class of Woodin cardinals, the theory ...

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### Mitchell, Steel. FSIT. Lemma 2.8: Is $k$-solidity actually needed?

Consider the following result (which is Lemma 2.8 in Mitchell and Steel's paper on Fine Structure and Iteration Trees):
Lemma 2.8 Let $\pi \colon \mathcal{H} \to \mathcal{M}$ be generalized $r \...

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### Strong determinacy principles for “small” sets

In the course of a (fun but silly) project I'm working on, I've run into the following class of determinacy notions:
Suppose I have a "reasonably definable" class $\mathcal{C}$ of games (in the usual ...

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### The reals in $L$

Assume "$0^\#$ exists".
We know that $0^\#$ is a $\Pi^1_2$-singleton. That means, there is a Shoenfield tree $S$ on $\omega \times (\omega \times \omega_1)$ so that $$x = 0^\# \leftrightarrow S_x \...

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### Core model with $\omega$ Woodin cardinals

In [KwoM] it is proved that the the core model K exists under the assumption that there is no inner model having a Woodin cardinal and satisfying ZFC. Furthermore, they also mention that the result is ...

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### Does absoluteness imply a club dichotomy?

My question is about two types of consequence of large cardinals, considered over ZFC on their own.
First, we have statements of the form, "The club filter on $\omega_1$ is an ultrafilter when ...

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### Natural length is a cardinal if it's a limit ordinal

Let E be a (Mitchell-Steel) extender over some M. Recall that the natural length of E, $\nu_E$, is the strict sup of the generators of E and $\kappa^{+M}$. It is claimed in both "Fine structure and ...

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### Breaking determinacy with forcing, and then fixing it

While forcing is usually presented over models of ZFC, it works equally well over models of ZF (or even less). However, the general theory of forcing becomes much stranger (much like the general ...

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### How verminous are mice?

Recall that a mouse is a structure of the form $(J_\alpha[U],\in,U)$ with $U$ being an amenable ultrafilter with some iterability properties.
One of the interesting facts about mice is that given two ...

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### Are generators of extenders cardinals?

Say $E$ is a $(\kappa,\lambda)$-extender on some model $\mathcal M$ of set theory, and consider the ultrapower embedding $i:\mathcal M\to\text{Ult}(\mathcal M,E)$. Now recall that a generator of $E$ ...

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### Preservation of Woodinness when it overlaps the active extender

I'm trying to show that if a premouse $\mathcal M$ is 1-small then it's also tame.
Definition. $\mathcal M$ is 1-small if for every extender $E$ on the $\mathcal M$-sequence, $\mathcal J^{\mathcal ...

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### Ordinal-definable witnesses to the perfect set property?

This possibly a very basic descriptive set-theory question; if it is too basic for MO, feel free to migrate.
Throughout we work in ZF+AD. My question is:
If $A$ is an uncountable OD set of reals, ...

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### On thin $\Sigma^1_2$ equivalence relations

This question is regarding Hjorth's paper "Some applications of coarse inner model theory", J. Symbolic Logic 62 (1997), no. 2, 337–365.
Hjorth claims that if $E$ is a thin $\Sigma^1_2$ equivalence ...

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### Showing that $\alpha$ isn't a cardinal in $J_{\alpha+1}^{\vec E}$ for a fine extender sequence $\vec E$

In [FSIT] and [OIMT] it is claimed that there is a surjection from $P(\kappa)\cap J^{\vec E}_{\nu(E_\alpha)}\times[\nu(E_\alpha)]^{<\omega}$ onto $\alpha$, and that this surjection lies in $J_{\...

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### Is it inconsistent for a model of set theory to contain its own first order theory?

I am wondering if it is inconsistent to have a model of set theory V such that V contains an $A\subset \omega$ that codes its first order theory.I.e. for all $\{\underline\epsilon\}$-sentences $\phi$, ...

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### Precipitous ideals and GCH

It is well known that ZFC + "There is a measurable cardinal" is equiconsistent with ZFC + "There is a precipitous ideal on $\omega_1$." Is ZFC + "There is a measurable $\kappa$ such that $2^\kappa &...

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### What is the motivation behind inner model theory?

Inner model theory aims to construct canonical inner models which captures as much of V as possible, which now is formulated more concretely as to build (fine structural) mice that contain many large ...

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### A Question on HOD, V and GCH

The theorem 1.1 of the following paper:
Mohammad Golshani, V, HOD, and the GCH, Journal of Symbolic Logic.
states that:
Theorem: Assume $V\models ZFC+GCH+~\text{There exists a}~(\kappa+4)-\...

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### Elementary chains in forcing extensions of $M_1$

Let $M_1$ be the canonical inner model with one Woodin cardinal $\delta$. Now suppose that $\mathbb{P}$ is a forcing notion of size $< \delta$, which preserves $\omega_1$ and that $G$ is a generic ...

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### Characterization of $L[T_{2n+1}]$ as a direct limit of mice

I am asking for a reference request/proof sketch for the result of Steel
that characterizes $L[T_{2n+1}]$ as a direct limit of mice.
Given that both $L[T_{2n+1}]$ and $M_{2n}$ have a $\Sigma_{2n+2}$ ...

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### Consistency strength of being strong cardinal and indestructible under collapses

What is the consistency strength of the following statement:
$\kappa$ is a strong cardinals and it is indestructible under $Col(\kappa, <\theta),$ where $\theta> \kappa$ is some fixed ...

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### Topological tameness beyond the Gandy-Harrington topology

The Gandy-Harrington topology on $\omega^\omega$ is the topology generated by all lightface $\Sigma^1_1$ sets; that is, all sets which are continuous-in-the-usual-sense images of $\omega^\omega$.
...

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### What consistency results follow the assumption: $\forall\alpha(\aleph_{\alpha+1}\nleq2^{\aleph_\alpha})$?

In a recent question on Math.SE it was asked whether or not For every infinite cardinal $\mathfrak m$ there is no $\aleph$ number, $\kappa$, such that $\mathfrak m<\kappa<2^{\mathfrak m}$.
By ...

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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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### Consistency strength of $\aleph_2$-Souslin hypothesis

Question 1. What is known about the consistency strength of $\aleph_2$-Souslin hypothesis?
Question 2. What is known about the consistency strength of having both $\aleph_2$-Souslin hypotheis and $\...

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### Core model for supercompact cardinals and iteration trees

I have a few somehow related questions:
Question 1. What do we expect for an inner model $\mathcal{K}$ to be a core model for a supercompact cardinal? What properties should it have, and what ...

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### Classifying set theories whose standard models sharing the same ordinals are equal

Let's say that a (recursively axiomatizable) set theory $T$ extending ZF is "ordinal-categorical" if, whenever $M$ and $N$ are standard models of $T$ sharing the same ordinals, one has $M = N$. For ...

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### Inner model in which every uncountable cardinal is large

The following is known:
$(*)$ If $0^\sharp$ exists, then any uncountable cardinal is is an inaccessible cardinal (and even more) in $L$.
My question is that:
Are there any large cardinal ...

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### Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)

Gödel's constructible universe ($L$) is defined using definable power set operator in first order logic ($\mathcal{L}_{\omega ,\omega}$). One can produce such a universe in infinitary logics in ...

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### What is known about equiconsistency of PFA and existence of supercompact cardinals?

Question: What is the last status of known partial results and new approaches on the problem of equiconsistency of PFA and existence of supercompact cardinals?

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### Latest stand of core model theory?

What is the "strongest" core model to this day? In particular, how far are we from a core model for supercompact cardinals? There are rumors of some notes from a workshop in 2004:
http://www.math.cmu....

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### What is the role of absoluteness in existence of a non-trivial self elementary embedding on an inner model?

ِDefinition (1): Call an inner model M "rigid" if there are no non-trivial elementary embeddings $j: M\longrightarrow M$.
It is possible that $L$ is not rigid, while the problem is open for $HOD$. ...