# 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.

72
questions

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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 ...

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### Inner models with all sets generic

Question: Under large cardinal axioms, what is the intersection of all inner models $M$ of ZFC such that every set in $V$ is set-generic over $M$?
Every set belongs to a generic extension of HOD, and ...

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### Absoluteness and the scale property for $Π^2_2$ or $Σ^2_2$

Under the diamond principle $◊$ and large cardinal axioms, which of the two pointclasses $Π^2_2$ or $Σ^2_2$ is expected to have the scale property?
Because conditional $Σ^2_2$ absoluteness under $◊$ ...

**3**

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178 views

### Pointwise definable models of determinacy

Suppose that the Axiom of Determinacy (AD) holds in $L(ℝ)$, and for some statement φ, $α$ is minimal such that $L_α(ℝ)⊨φ$. Do definable (in $L_α(ℝ)$) elements of $L_α(ℝ)$ form an elementary ...

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### Does $Π^1_{2n+1} = (Σ^1_{2n+1})^{M_{2n-1}}$?

Every $Π^1_1$ formula $φ$ without free second order variables can be converted into a $Σ^1_1$ $ψ$ such that $φ ⇔ ψ^\mathrm{HYP}$, and vice versa. ($\mathrm{HYP}$ is the hyperarithmetical universe, ...

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### $\lt_{ip}$ is a well-defined well-ordering of iterable set premice

I am cross-posting this question from MSE, where I asked it about $3$ months ago and I decided to ask it here as well.
This question of mine arises from Kanamori's the higher infinite, where he tries ...

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**1**answer

195 views

### Uniqueness of countable version of $L[U]$?

Suppose $\alpha$ is a countable ordinal and $U_0,U_1,\kappa$ are such that $L_\alpha[U_i] \models \mathrm{ZFC} + U_i$ is a normal ultrafilter on $\kappa$. Does $U_0 = U_1$?
The argument for ...

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**1**answer

429 views

### What happens with large singular cardinals on the far side of the HOD dichotomy?

Woodin's HOD Dichotomy Theorem says that if an extendible cardinal exists, then either $V$ and $HOD$ are rather close or rather far apart. My question is whether the "far" case can be strengthened in ...

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345 views

### What are some good lower bounds on the consistency of the failure of the PCF conjecture?

Shelah's celebrated theorem states that $\aleph_\omega$ is a strong limit cardinal, then $2^{\aleph_\omega}<\aleph_{\omega_4}$.
But the conjecture is that $\omega_4$ can be provably replaced by $\...

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174 views

### Symmetry between V and HOD

Is it consistent that the set of ordinal definable real numbers is countable, but for every $y∈\mathrm{OD}∩ℝ$, every true $Σ_2^{V,y}$ statement holds in $\mathrm{HOD}$?
Note that $Σ_2^V$ is the best ...

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538 views

### Does inner model theory seek canonical models for large cardinals?

Like the author of this question, I have heard that a main goal of inner model theory is building canonical inner models for large cardinals. My questions are: (a) Is this accurate? (b) If so, in ...

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65 views

### 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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216 views

### 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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210 views

### 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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311 views

### 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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694 views

### 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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224 views

### 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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267 views

### 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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4k views

### 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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331 views

### 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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195 views

### 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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199 views

### 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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283 views

### 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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355 views

### 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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371 views

### 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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216 views

### 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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275 views

### 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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254 views

### 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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862 views

### 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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375 views

### 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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267 views

### 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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351 views

### 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 ...