# 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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### Inner model of "CH + large cardinals" that satisfies MM?

I've been told that if $V$ has CH and large cardinals, then there is an inner model in which MM holds. The sketch is as follows:
Any $\Sigma^2_1$ sentence that can be forced is already true in such $V$...

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### In the construction of $K^c$ for sequences of measures, why do we only add measures of cofinality $\omega_1$?

$\require{cancel}$
I have a question about the definition of $K^c$ for sequences of measures in Mitchell's "The Covering Lemma" chapter in the Handbook. I will give the definition he gives ...

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### Weak extender models for supercompactness without choice

Assume ZFC and a supercompact cardinal $κ$. Is it consistent that there is a weak extender model $N⊨\text{ZF}$ for supercompactness of $κ$ such that the axiom of choice and well-ordering of $P_κ(λ)^N$...

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### Do precipitous ideals "always" come from collapsing?

It's well-known that if $\kappa$ is a measurable cardinal, then there is a poset $\mathbb{P}$ that forces $\kappa$ to carry a precipitous ideal.
Suppose that $\omega_1$ carries a preciptous ideal $I$.
...

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### Friedman's proof of covering lemma for $L$

There is a two-page proof of the covering lemma for $L$ using $\Sigma_n$ substructures (Theorem 3.10) in Sy Friedman's Fine Structure and Class Forcing, compared to the proof that spans about twenty ...

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### The strongest reflection principle that does not violate covering lemmas

#-generated reflection, or Indiscernible-generation, is considered to be the strongest reflection principle that does not violate the covering lemma in L. [1]
Is there a way to extend this success to ...

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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 ...

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### Precipitous ideal and inner model

Assume $\kappa$ is measurable, $U$ is its unique normal measure, $V=L[U]$. We levy collapse $\kappa$ to make it become $\omega_1$.
If we don't have the inner model condition, then we only know that $\...

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### What's the definition of a mouse in Mitchell's handbook article "the covering lemma"?

In the book "handbook of set theory", in the chapter "the covering lemma", definition 3.24, Mitchell defines what is mouse.
However he did not give any definition of $\mathcal{U}_\...

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### The most powerful inner model and a $\Delta^2_1$ well-ordering of the reals

With the current research, it seems that we are in a position to get extremely powerful absoluteness theorems (like $\Sigma^2_0$-absoluteness, $\Sigma^2_1$-absoluteness, $\Sigma^2_2$, $\diamondsuit_G$,...

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### Inner model for KP and a Well-Ordering of the Reals

It is well known that Gödel proved the following theorem:
$\mathsf{ZFC + V=L}$ has a $\mathit{\Delta}^1_2$-good well-ordering of $\mathbb{R}$. (Gödel, Addison)
So:
Is there an inner model for KP/Z/....

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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### 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 ...

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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 ...

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### 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. ...

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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 ...

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### 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\...

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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 $\mathsf{FOL}$ ...

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### 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 $\...

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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 ...

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### 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 $...

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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) \\
\...

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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}(\...

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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 ...

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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 ...

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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 ...

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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 ...

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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 $A\not\in L$ ...

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### 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(...

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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 ...

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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 ...

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### 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 ...

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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?

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### 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. ...

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### 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 \...

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### 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}^{\...

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### 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 ...

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### 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 $...

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### 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 ...

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### 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 ...

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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 $◊$ ...

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### 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 ...