# Questions tagged [large-cardinals]

The large-cardinals tag has no usage guidance.

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### What's the consistency strength of this kind of cardinal?

My friend introduced the following notion:
Let $\xi > 0$, $\eta$ be ordinals, $n$ be a natural number and $\mathcal{A}, X$ be classes. A cardinal $\kappa$ is called $\mathcal{A}\textrm{-}\eta\...

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### Is this strengthening of the definition of weakly Shelah cardinals equivalent to being weakly Shelah?

This MathOverflow question by Trevor Wilson defines weakly Shelah cardinals as follows:
A cardinal $\kappa$ is weakly Shelah if for all $f : \kappa \to \kappa$ there is some $\alpha < \kappa$ that ...

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### Reinhardt's ultimate classes

In the preface to Sets and Classes by Muller, several research programs are outlined that were in development concurrently with publication (or finished slightly beforehand) that he would have liked ...

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### Why may club Berkeley cardinals not be Berkeley?

"Large Cardinals beyond Choice" makes the following definitions:
$\delta$ is a Berkeley cardinal if for every transitive set $M$ such that $\delta \in m$ and every $\eta \lt \delta$ there ...

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### What's the consistency strength of this strengthening of weakly superstrong cardinals?

Recall that a cardinal $\kappa$ is weakly superstrong if, for every $A \subseteq V_\kappa$, there is a cardinal $\lambda$ and a set $A^* \subseteq V_\lambda$ such that $\langle V_\kappa, \in, A \...

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### Thinning chains of elementary extensions

I'm bumping this question, since I'm still curious regarding the answer but this question seems to have gone unnoticed.
This is a follow-up to this question, regarding a stronger variant of ...

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### What's the consistency status/strength of this limitation principle?

$\DeclareMathOperator\iCard{iCard}$In a prior posting If we limit matters what ZFC can prove, would that be consistent? to MO, I tried to capture the informal principle of whatever ZFC proves, it is, ...

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### Equivalences of $\mathcal{F}$-Mahloness

Taken from Math Stack Exchange.
Let $\mathcal{F}$ be a set of $\mathcal{L}_\in$-formulae, $\kappa$ be a cardinal and $A \subset \textrm{Ord}$. Then, $\kappa$ is called $\mathcal{F}$-Mahlo if $A \cap \...

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### Are those two theories equivalent?

Lets denote any set that is "the set of all strictly smaller subsets of it" as size- unreachable. Formally: $$\operatorname {size-unreachable}(X) \iff \\\forall Y (Y \in X \iff Y \subset X \...

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### Shelah's "Can you take Solovay's inaccessible away?"

I was wandering if there was a book, thesis or some notes where Shelah's argument for
$\mathtt{ZF}+\mathtt{DC}+$"All sets of reals are Lebesgue measurable" is equiconsistent with $\mathtt{...

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### $\mathtt{PSP}$ implies the consistency of inaccessible cardinals

I'm looking for the proof that $\mathtt{PSP}$, the statement that every uncountable subset of the the Baire space $\mathbb{N}^\mathbb{N}$ contains an homeomorphic copy of the Cantor space $2^\mathbb{N}...

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### What is the most "Icarus" Icarus set axiom?

We call a set $X ⊆ V_{λ+1}$ an Icarus set if there is an elementary embedding $j : L(X, V_{λ+1}) ≺ L(X, V_{λ+1})$ with $\mathrm{crit}(j)< λ$.
But this raises the question: What is the most "...

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### Why does the second smallest worldly cardinal believe the smallest worldly cardinal is worldly?

It is known that the property of being a worldly cardinal is not absolute (a cardinal $\kappa$ is worldly iff $V_{\kappa} \vDash \textsf{ZFC}$). See here and here for more.
This said, it is the case ...

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### Can Reinhardt cardinals be compatible with Choice in absence of Extensionality?

Is the proof of existence of Reinhardt (and higher) cardinals violating Choice dependent on Extensionality in an essential manner?
What I mean is if we work in $\sf ZFA$ would it be possible to have a ...

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### Supercompact cardinal above a measurable and fixed points of the ultrapower map

Let $\kappa$ be a measurable cardinal and let $j:V\to M$ be the ultrapower map. Assume $\mu$ is a supercompact cardinal with $\mu>\kappa$. What can we say about $j(\mu)$? Is it true that $j(\mu)=\...

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### Are there interesting examples of theorems proved using ‘height’ extensions?

It's well known that forcing is more than a tool for proving independence: We can prove theorems and formulate axioms in theories like $\mathsf{ZFC}$ by moving to forcing extensions (e.g. $\mathfrak{p}...

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### Can the essence of the $0^\#$ LCA be weakened to an axiom not requiring uncountable cardinals?

"$0^\#$ exists" is an informally stated large cardinal axiom that is to be understood as "there is an uncountable set of Silver indiscernibles", "every uncountable cardinal is ...

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### How could we define "recursively greatly Mahlos"?

A common action in set theory is making a large cardinal axiom "recursive", i.e. turning it from a large uncountable cardinal to a large countable ordinal. For example:
Recursively regular =...

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### How to characterize properties that behave well with Reflection Principles

I'm interested in Reflection Principles but I can't find any references of works around criteria to classify properties well-behaved relatively to reflection, or at least features that properties must ...

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### What are the known large cardinal axioms for which weaker and stronger set theories "catch up"?

I will clarify what I mean by the title in the following four ways:
For which cardinals $\kappa$ do we have that ZFC-(Powerset axiom)+$\exists\kappa$ is equiconsistent with ZFC? If that is not ...

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### Can local $0^\#$ exists in L?

Assume $0^\#$ exists and there is an inaccessible cardinal.
Are there two transitive sets $M,N$ s.t. $M\in N,M\vDash ZF+V=L[0^\#],N\vDash ZF+V=L$?

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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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### How far $HOD_{A}(X)$ can be from $V$ if exist choiceless cardinals?

If exist totaly Reinhardt or Berkeley cardinal then $V≠HOD_{A}(X)$, where $A$ predicate defined in some class.
If exist 0# then any uncountable cardinal large in $L$
$V$ also can be far from $HOD$, ...

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### $\aleph_1$-complete fine measures on $P_\kappa(\lambda)$

Definition A fine measure on $P_\kappa(\lambda)$ is a non-principal ultrafilter on $P_\kappa(\lambda)$ which contains all upper cones $\uparrow{x}=\{y\in P_\kappa(\lambda)|x\subset y\}$, for all $x\...

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### If $L_\alpha \vDash ZFC$, then do we have $L_{\alpha+1} \vDash \alpha\text{ is inaccessible}$?

Here we choose the definition of "is a cardinal" as there is no surjective map from a smaller ordinal to it.
It's easy to prove that, if $L_{\alpha+1} \vDash\ \alpha\text{ is inaccessible}$, ...

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### Stronger (?) form of Vopenka's principle

A category $\mathcal{C}$ is called $\textbf{discrete}$ if the only morphisms are identity morphisms.
Consider the following weaker notion: a category $\mathcal{C}$ is called $\textbf{totally ...

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### Measurable cardinals from saturated ideals

Assume that $\omega<\kappa_1<\dotsb< \kappa_n$ are infinite cardinals such that for each $1\le i\le n$ there is a $\kappa_i$-complete, $\kappa_i^+$-saturated ideal $\mathcal I_i\subset \...

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### Consistency strength about Ramsey M-rank and Mahlo-Ramsey cardinal

In the website "Cantor's attic", there are a long list of large cardinal axioms arranged by consistency strength.
In the list, "α-Mahlo Ramsey" is placed higher than "Ramsey M-...

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### How big a "scaffold" does second-order logic need to detect its own equivalence notion?

(Previously asked and bountied at MSE:)
Let $\Sigma$ be the language consisting of a single binary relation symbol. Second-order logic can "detect" second-order-elementary-equivalence of $\...

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### 'Maximising interpretative power entails maximising consistency strength'?

I'm hoping there is a clear mathematical answer to this question (hence asking it here) rather than anything more exegetical (in which case it's presumably not appropriate for this site).
In his paper ...

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### Consistency strength of Sy Friedman's result about admissibility spectrum

A result by Sy Friedman in his book "fine structure and class forcing", is that, assume $0^\sharp$ exists, there exists a real number R such that the ordinals admissible in R (called $\...

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### How large is the supremum of minimal $V$-heights of all first-order set theories formulated in a particular language of FOST?

Fix a language $\mathcal{L}$ of first-order set theory. For this question, we can assume that $\mathcal{L}$ is the language described in Chapter 1 of “An introduction to set theory” [William A. R. ...

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### What is known about the least cardinal where $\kappa$ fails to be supercompact?

Assume $\kappa$ is $\lambda$-supercompact for some $\lambda$ but not fully supercompact. Are there any known restrictions (or provably non-restrictions) on the least $\delta$ such that $\kappa$ is not ...

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### An inconsistency about Magidor models

Recall that a model $M\prec V_\theta$ is $\kappa$-Magidor if it has transitive intersection with $\kappa$ and its transitive collapse is equal to $V_\alpha$ for some $\alpha<\kappa$. The ...

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### Can HCD accommodate all known large cardinal axioms?

HOD has been found to be useful in that it is an inner model that can accommodate essentially all known large cardinals.
However, there is a definable well ordering over HOD, so it cannot satisfy ...

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### Dehornoy's proof that the application of two elementary embeddings is an elementary embedding

What is meant by the statement and the proof of Lemma 3.2 in Chapter XII of Dehornoy's book Braids and Self-Distributivity?
That lemma states "Assume that $j_1$ and $j_2$ are elementary ...

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### Motivation for Laver's use of large cardinals to show finite combinatorial properties of Laver tables

Laver showed in 1995 that the period of the first row of certain Laver tables is unbounded, assuming that a rank-into-rank cardinal exists.
The most accessible proof of his result that I was able to ...

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### A restricted form of the inner model hypothesis

Previously asked and bountied at MSE, with slight difference. To keep things relatively simple I'm presenting a somewhat-butchered version of the IMH; for more details, see S.-D. Friedman, Internal ...

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### Are the following two "tree properties" equivalent?

Let $\kappa$ and $\lambda$ be cardinals. A thin $(\kappa,\lambda)$-list is a function $L:[\lambda]^{<\kappa}\longrightarrow [\lambda]^{<\kappa}$ such that for all $x\in[\lambda]^{<\kappa}$, $...

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### A proper class of ordinals without an infinite constructible subset

If $0^\sharp$ exists then the $L$-indiscernibles form a proper class of ordinals without any infinite constructible subset, as $0^\sharp$ can be defined from any infinite increasing sequence $\langle \...

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### Countably closed end-extensions of elementary submodels

The following is well-known. If $\kappa$ is measurable, $\theta > \kappa$, and $M \prec V_\theta$ has size $<\kappa$, then there is $N\prec V_\theta$ such that $N \supseteq M$, $M \cap \kappa \...

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### Compatibility of Łośian phenomena in second-order logic

(Throughout, all ultrafilters are nonprincipal.)
Given a property $P$ - really, a sentence in some appropriate logic - say that a ultrafilter $\mathcal{U}$ on a cardinal $\kappa$ averages $P$ iff for ...

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### Can a Vopenka cardinal be supercompact?

Can a Vopenka cardinal be supercompact?
I asked a weaker question on here before. Unfortunately, I don't know enough set theory to see whether the positive answer there generalizes to a positive ...

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### Weak compactness is to trees as [?] is to lattices?

Let $\kappa$ be an inaccessible cardinal. Recall that $\kappa$ is weakly compact if every tree of height $\kappa$ has either a level of size $\kappa$ or a branch of size $\kappa$.
So if $\kappa$ is a ...

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### $n$-ineffable and $n$-Ramsey hierarchies

The paper Games and Ramsey-like cardinals by Nielsen and Welch 2018 defines $n$-Ramsey cardinals as follows (this is not quite the same definition but it's equivalent): $\kappa$ is $n-1$-Ramsey if ...

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### Finite axiomatizability of $\mathrm{WA}_0$

$\mathrm{WA}_0$ (which belongs to a hierarchy of theories called the wholeness axioms) is a theory extending ZFC where there is an elementary embedding $j:V \to V$ which is non-trivial and amenable, ...

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### Extendible and enhanced supercompact cardinals

The paper "The large cardinals between supercompact and almost-huge" (2013) by Norman Perlmutter makes the following definition:
A cardinal $\kappa$ is enhanced supercompact if and only if ...

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### Three questions from Kentaro Sato's paper about the n-fold large cardinal hierarchy

In the paper Double helix in large large cardinals and iteration of elementary embeddings
there are three things mentioned as unknown which I can answer:
[S]everal results known for ordinary ...

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### Class-theoretic sentences that are $\Pi^1_1$ or $\Pi^1_2$

I'm looking for the following:
(1) an example of a $\Pi^1_1$ class-theoretic sentence that has no known equivalence to a $\Sigma^1_1$ sentence, even if large cardinal hypotheses or reflection ...

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### Which step is wrong in the following simplification of Silver's forcing?

Theorem: If M is a countable transitive model of ZFC, and $\kappa$ is a supercompact cardinal in M, and $2^\kappa=\kappa^+$. Then there exists a forcing extension M[G] such that $\kappa$ becomes a ...