Questions tagged [large-cardinals]
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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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$\aleph_1$-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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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 ...
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Possible inconsistency of weakly Shelah cardinals (I hope not)
A 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 is ...
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Is Vopenka's Principle + "ORD has the tree property" consistent?
Vopenka's principle implies the existence of weakly compact cardinals (a proper class of them, I believe). My question is whether Vopenka's principle is consistent with the assertion that the universe ...
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Is the supremum of L-definable cardinals silver-indiscernible
Let $\kappa$ be the supremum of ordinals first order definable in L without parameters. Assume $0^\sharp$ exists. Is $\kappa$ the least silver indiscernible ordinal?
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Elementary embeddings and replacement
Let $\alpha\not= 0$ be such that for every $\beta<\alpha$ there is $\beta<\gamma<\alpha$, where $V_\gamma$ is an elementary substructure of $V_\alpha$. In other words, $V_\alpha$ is a limit ...
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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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Compactness number for a fragment of second-order logic
Previously asked and bountied without response at MSE.
This question is a companion to this one, about a tame(?) fragment of second-order logic with the standard semantics, $\mathsf{SOL}$, motivated ...
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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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Analogues of worldly cardinals for (an unusual version of) second-order $\mathsf{ZFC}$
Let $\mathfrak{ZFC}(\mathsf{SOL})$ be the theory in second-order logic (with the standard semantics) gotten from $\mathsf{ZFC}$ by modifying the Separation and Replacement schemes to apply to ...
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Determinacy of symmetric games
Is it consistent that for all ordinals $α$ and $λ$ and infinite regular cardinals $κ$, games on $V_λ$ with game length $κα$ and $\mathrm{OD}(\mathrm{On}^κ)$ payoff that depends only on the set of all ...
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Cardinality of infinite towers of Alephs - can tower be more than countable?
Lets define function T as
$$T(0) = \aleph_0$$
$$T(1) = \aleph_{\aleph_0}$$
$$T(2) = \aleph_{\aleph_{\aleph_0}}$$
etc
No finite tower of alephs can reach the first inaccessible cardinal
My questions ...
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Complexity of $Σ_n$ theory of $H(ω_2)$ under MM$^{++}$
Under Woodin's $\mathbb{P}_\text{max}$ axiom (which is implied by MM$^{++}$), what is the complexity of the $Σ_n$ theory of $H(ω_2)$? Same question for $Σ_n(I_\text{NS})$ (i.e. using the ...
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Proper class of nested rank into rank embeddings
I propose the following large cardinal axiom:
There exists a proper class of cardinals $\lambda$, such that for each $\lambda$, there exists a rank-into-rank embedding $j: V_\lambda \rightarrow V_\...
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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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Large cardinals without replacement
Let $ZC$ be Zermelo set theory with choice, which differs from $ZFC$ in omitting the axiom scheme of replacement. EDIT: I think I want to include foundation in the axioms, which apparently isn't ...