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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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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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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Consistency strength of saturated ideals on $[\lambda]^{<\kappa}$
In the Handbook of Set Theory, Foreman notes that in models constructed by Magidor from a huge cardinal, there exists a normal ideal on $[\omega_2]^{<\omega_1}$ that is $\omega_3$-saturated. Other ...
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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 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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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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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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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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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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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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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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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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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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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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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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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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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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"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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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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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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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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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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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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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 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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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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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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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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"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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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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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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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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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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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 $S$ ...