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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What is the exact consistency strength, if known, of the precipitousness of $I_{\text{NS}}$ on a successor of a singular cardinal $\kappa$?

The question is in the title. Jech states in his book on page 696 that the consistency strength is "in the region of Woodin cardinals," which is frustratingly imprecise. I tried to find a ...
Connor W's user avatar
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Destroying the iterability of $M_1^\#$

Suppose $M_1^\#$ exists and is $\omega_1$-iterable. Is it consistent that we can go to a generic extension $V[G]$ where $M_1^\#$ is no longer $\omega_1$-iterable? Or "worse" $M_1^\#$ is no ...
Toby Meadows's user avatar
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What is the evidence for and against the HOD conjecture?

I'm aware that the HOD conjecture is implied by the Ultimate-L conjecture, but I don't know what the evidence is for the Ultimate-L conjecture. On the other hand, I'm aware the evidence against the ...
Someone211's user avatar
3 votes
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Universe V = Ultimate L inside set theoretic multiverse

Good day to you all, I would like to ask a question about relation between Prof. H. Woodin V = Ultimate L and a concept of set theoretical multiverse as proposed by Prof. Hamkins. If V = Ultimate L ...
Pan Mrož's user avatar
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Inner model with a $\mathit{\Delta}^1_3$-good well-ordering of the reals

The constructible universe $L$ has some nice properties: $L$ has a $\mathit{\Delta}^1_2$-good well-ordering of $\mathbb{R}$. (Gödel, Addison) For any $\mathit{\Sigma}^1_2$ formula $\varphi(x)$ and a ...
Lorenzo's user avatar
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Is Jensen's covering lemma meaningful in a platonist's view?

The typical applications of fine structure theory are finding out the lower bounds of consistency strength of axiom systems. In such a proccess, we also constructs many combinatorial objects in core ...
Reflecting_Ordinal's user avatar
5 votes
1 answer
170 views

Are there premice that are $\omega_1$-iterable but not $(\omega_1+1)$-iterable?

For (hopefully) simplicity, let a premouse be defined coarsely as in Martin and Steel's 1994 paper, Iteration Trees. Is (or is it consistent that) there is a premouse that is $\omega_1$-iterable but ...
Toby Meadows's user avatar
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Does $\mathit{Aut}(\mathbb{R};+)$ have a copy in $L(\mathbb{R})$ granting large cardinals?

Throughout, work in $\mathsf{ZFC}$ + large cardinals (let's say a proper class of Woodin limits of Woodins but I'm happy to go higher if that would help). Let $\mathcal{R}=(\mathbb{R};+)$ be the ...
Noah Schweber's user avatar
17 votes
3 answers
908 views

Minimum transitive models and V=L

Is there a c.e. theory $T⊢\text{ZFC}$ in the language of set theory such that the minimum transitive model of $T$ exists but does not satisfy $V=L$? You may assume that ZFC has transitive models. ...
Dmytro Taranovsky's user avatar
2 votes
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"Very $L$-like" models, part 2: combinatorics

Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ and having the finite use property and the strong downward Lowenheim-Skolem property together with, for each finite ...
Noah Schweber's user avatar
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300 views

Is there a "nice" inner model for $\mathsf{ZF}$ + a Dedekind-finite infinite set of reals?

Below, given a formula $\varphi$ which $\mathsf{ZF}$ proves defines a set of reals and an inner model $W$, I'll write "$\varphi^W$" and "$L(\varphi^W)$" for "$\{x:W\models\...
Noah Schweber's user avatar
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0 answers
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"Very $L$-like" models, part 1: large cardinals

(The original version of this question was much narrower and less natural; but see the edit history if interested.) Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ ...
Noah Schweber's user avatar
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205 views

Absoluteness of the core model under a proper class of completely Jónsson cardinals

Example 2.4.2 of Larson's The stationary tower (based on Woodin's lecture) describes how we can absorb a generic filter over a set forcing in a definable inner model into a generic extension by $\...
Hanul Jeon's user avatar
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Intuition for branch uniqueness in inner model theory

In inner model theory, what is the intuition behind the expectation that under appropriate conditions, we should have a single preferred branch to continue an iteration at a limit stage? At the level ...
Dmytro Taranovsky's user avatar
2 votes
0 answers
120 views

Inner model theory using indiscernibles

Has an inner model theory been developed on the basis of indiscernibles rather than measures? Is there a reasonable formalization at the level of overlapping extenders? Fine-structural models beyond $...
Dmytro Taranovsky's user avatar
3 votes
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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
1 answer
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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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Is there a relationship between the $\Omega$-conjecture and choiceless large cardinals?

Woodin’s $\Omega$-conjecture is absolute under set-forcing. One proposal to decide it is that some large cardinal hypothesis might refute it. On the other hand, Woodin is seeking extensions of the ...
Monroe Eskew's user avatar
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Literature on the reals or their gaps in $L[0^\sharp]$?

I'm doing my Bachelor's Thesis on the continuum in $L$ and $L[0^\sharp]$. In $L$ I study the gaps without new reals (sets of natural numbers) in the hierarchy, as presented in Gaps in the ...
Martín S's user avatar
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Recent literature on the gaps of reals on $L$ or other inner models?

I'm doing my Bachelor's Thesis on Gödel's constructible universe $L$. I'm interested in the gaps without new reals (sets of natural numbers) in the hierarchy, as presented in Gaps in the constructible ...
Martín S's user avatar
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A question about Shoenfield's absoluteness theorem and $0^{\sharp}$

Shoenfield's theorem states that any $\Pi_{3}^{1}$ sentence that holds in $V$ holds in $L$, but I know that it is consistent with ZFC that there exists a set $A\subset \mathbb{N}$ where $A\not\in L$ ...
Ândson josé's user avatar
6 votes
0 answers
225 views

Are initial segments of coherent measure sequences coherent?

This question is about the "old-fashioned" coherent sequences, in the style of Mitchell Mitchell, W. (1974). Sets constructible from sequences of ultrafilters. Journal of Symbolic Logic, 39(...
Miha Habič's user avatar
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11 votes
1 answer
366 views

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
226 views

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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The core model and elementary embeddings

Let $K$ be the core model (below a Woodin cardinal). Let $j \colon K \to M$ be an elementary embedding, where $M$ is well founded. Under which conditions can we conclude that $j$ is an iterated ...
Yair Hayut's user avatar
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3 votes
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199 views

Independence through forcing vs generic collapses

Are there known statements in $V_{ω+ω}$ independent through forcing after $\mathrm{Col}(ω,<κ_1)*\mathrm{Col}(κ_1,<κ_2)*\mathrm{Col}(κ_2,<κ_3)*...$ where $κ_1<κ_2<κ_3<...$ are ...
Dmytro Taranovsky's user avatar
8 votes
1 answer
286 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?
Someone211's user avatar
8 votes
0 answers
273 views

Inner models from highly saturated ideals

Solovay proved that a measurable cardinal is equiconsistent with the existence of an extension of Lebesgue measure to a full measure on $P(\mathbb R)$. The inner model direction is relatively simple. ...
Monroe Eskew's user avatar
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12 votes
1 answer
330 views

Getting a model of $\mathsf{ZFC}$ that fails to nicely cover an inner model

Consider the following statement: $(\dagger)$ $\ $ There is an inner model $M$ such that $M \models \mathsf{GCH}+\square$ and for every countable $X \subseteq \mathrm{Ord}$, there is a countable $Y \...
Will Brian's user avatar
3 votes
1 answer
304 views

If we have a class like $L$ but allowing a set number of unbounded quantifiers, is it strict superset of $L$?

The definition of $L$ only permits bounded quantifiers. If we allow a certain number of unbounded quantifiers, does this result in a strict superset of $L$? For example: $$ \operatorname{Def}^{\...
PyRulez's user avatar
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17 votes
2 answers
1k views

A “paradox” about the inner model problem

As stated in Woodin, Davis, and Rodriguez - The HOD dichotomy, a longstanding open problem in set theory is to construct a canonical inner model for supercompactness. In general there are various ...
Monroe Eskew's user avatar
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17 votes
1 answer
706 views

What sets can be unraveled?

A set $X\subseteq\omega^\omega$ is unravelable iff there is a possibly larger set $A$ and a clopen set $Y\subseteq A^\omega$ (with respect to the product topology coming from the discrete topology on $...
Noah Schweber's user avatar
5 votes
1 answer
260 views

Consistency strength of lifting through a lot of collapsing

What is the consistency strength of the following situation? $j : V \to M$ is an elementary embedding definable from parameters in $V$, with critical point $\kappa$. $\mathbb P$ is a forcing that ...
Monroe Eskew's user avatar
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5 votes
1 answer
370 views

GCH implies acceptability

I have been studying the concept of acceptability, particularly in its relation to GCH. There are many versions of it in the sources I have found, with some slight variations, and some of them are ...
Rodrigo Freire's user avatar
10 votes
0 answers
266 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 ...
Asaf Karagila's user avatar
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5 votes
0 answers
298 views

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 ...
Dmytro Taranovsky's user avatar
5 votes
0 answers
230 views

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 $◊$ ...
Dmytro Taranovsky's user avatar
3 votes
1 answer
279 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 ...
Dmytro Taranovsky's user avatar
11 votes
1 answer
418 views

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, ...
Dmytro Taranovsky's user avatar
7 votes
1 answer
261 views

$\lt_{ip}$ 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 ...
Shervin Sorouri's user avatar
5 votes
1 answer
236 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 ...
Monroe Eskew's user avatar
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7 votes
1 answer
550 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 ...
Monroe Eskew's user avatar
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13 votes
1 answer
417 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 $\...
Asaf Karagila's user avatar
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5 votes
0 answers
259 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 ...
Dmytro Taranovsky's user avatar
16 votes
1 answer
844 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 ...
Monroe Eskew's user avatar
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1 vote
0 answers
94 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 $S$ ...
Dmytro Taranovsky's user avatar
8 votes
0 answers
231 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 ...
Monroe Eskew's user avatar
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6 votes
0 answers
235 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}...
Stefan Mesken's user avatar
18 votes
1 answer
2k views

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 ...
Morteza Azad's user avatar