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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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 ...
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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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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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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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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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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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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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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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