Questions tagged [large-cardinals]
The large-cardinals tag has no usage guidance.
197
questions with no upvoted or accepted answers
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Supercompact and Reinhardt cardinals without choice
A friend of mine and I ran into the following question while reading about proper forcing, and have been unable to resolve it:
Definition. A cardinal $\kappa$ is supercompact if for all ordinals $\...
18
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899
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Souslin trees and weakly compact cardinals
In Souslin trees on the first inaccessible cardinal it is asked if it is consistent that there are no $\kappa-$Souslin trees at the least inaccessible cardinal $\kappa$.
In this question I would like ...
17
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755
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Ideas behind Gitik's solution of PCF conjecture
Recently Moti Gitik has refuted Shelah's PCF conjecture (see Short extenders forcings II ) by proving the following theorem:
Theorem. Assuming the consistency of infinitely many strong cardinals, one ...
16
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617
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Consistency strength of $j:L_δ→L_δ$ for some δ
What is the consistency strength of existence of a nontrivial elementary embedding $j:L_δ→L_δ$ for some ordinal $δ$?
The consistency strength is strictly between totally ineffable and $ω$-Erdős ...
14
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515
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Seeing what gets Harvey Friedman's "tangible incompleteness" principles into large cardinal territory
I'm trying to wrap my head around some of Harvey Friedman's recent, unpublished work on his tangible incompleteness project, and I'm trying to see the link between his "tangible statements" (...
14
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342
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The failure of GCH al $\aleph_\omega$ by nice forcing
There are several ways to force $GCH$ below $\aleph_\omega$ and $2^{\aleph_\omega}> \aleph_{\omega+1},$ say:
1) The Gitik-Magidor's extender based forcing, see Prikry type Forcings.
2) Woodin's ...
13
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0
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519
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Large cardinals arising from alternate set theories
My question is whether there are model-theoretic large cardinal axioms associated with $NFU$ - or more generally, with set theories other than $ZFC$.
Large cardinal properties generally come in one ...
13
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455
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Adding a saturated ideal
Is it consistent that there is no $\omega_2$-saturated ideal on $\omega_1$, but one is introduced by an $\omega_2$-closed forcing?
Some motivation:
If $\delta$ is a Woodin cardinal, then it remains ...
12
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343
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Can Friedman's property fail at or above a supercompact cardinal?
If $\kappa>\omega_1$ is a regular cardinal, $FP_\kappa$ is the assertion that every stationary subset of $\kappa$ consisting of ordinals of countable cofinality has a closed subset of order type $\...
12
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Does $2^{\aleph_0}\rightarrow [\aleph_1]^2_3$ require that the continuum is weakly inaccessible?
A classic result of Sierpiński shows that $2^{\aleph_0}\nrightarrow [\aleph_1]^2_2$, that is, there is a coloring of pairs of real numbers using two colors such that both colors appear on any ...
12
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361
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Singular Jonsson cardinals
Is the following consistent?
$(*)$: There exists a singular cardinal $\kappa$ such that :
(1) $\kappa$ is a Jonsson cardinal,
(2) $\kappa$ is not a fixed point of the $\aleph-$function, i.e., $\kappa ...
12
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Some questions about $0^{\sharp}$ and forcing over $L$
1-Let $P\in L$ be a nontrivial set forcing or even a tame class forcing (tameness in the sense of Sy Friedman; see for example his Handbook paper), and let $G$ be $P-$generic over $L$. Then it is well-...
11
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453
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$\Sigma^2_1$ and the Continuum Hypothesis
This is a follow up to Will Brian's answer to this recent question. In particular, quoting Brian:
"In fact, Paul Larson has pointed out to me that the statement "$\phi$ and $\phi^{-1}$ are conjugate"...
11
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290
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Preserving Jonsson cardinals
I am (still) interested in trying to characterize and describe forcings that preserve Jonsson cardinals. A cardinal $\kappa$ is a Jonsson cardinal if there is no Jonsson algebra on $\kappa$, i.e. ...
11
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Does Sageev's result need an inaccessible?
In 1981, building on work by Ellentuck in 1974, Sageev showed ("A model of ZF + there exists an inaccessible, in which the Dedekind cardinals constitute a natural non-standard model of arithmetic," ...
11
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Absoluteness of "$\kappa$-homogeneously Suslin" for sets of reals
What is known about the absoluteness, or lack thereof, of the notion of "$\kappa$-homogeneously Suslin" for sets of reals?
For example, if $A$ is $\kappa$-homogeneously Suslin and $\lambda > \...
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Is $\kappa \rightarrow [\kappa]^2_3$ the same as $\kappa \rightarrow [\kappa]^2_2$ for inaccessible $\kappa$
The principle $\kappa \rightarrow [\kappa]^2_\alpha$ states that whenever we have a coloring $c:[\kappa]^2\rightarrow \alpha$ there is $H \subset \kappa$ of size $\kappa$ s.t. $|c"[H]^2|<\alpha$.
...
10
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304
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Feferman's universes for proof assistants?
This question was prompted by a discussion from another MO question about the consistency of ZFC. There are some mathematicians who are comfortable with ZFC but uneasy with large cardinals. For them,...
10
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stationary reflection in $[\kappa]^\omega$
It is well-known that the following reflection principle is consistent relative to a supercompact:
For all $\kappa \geq \omega_2$ and all stationary $S \subseteq [\kappa]^\omega$, there is $X \...
10
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267
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Strongly compact vs Shelah cardinals
Does Con(ZFC+there exists a strongly compact cardinal) imply the Con(ZFC+there exists a Shelah cardinal)?
10
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356
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Inner models and strongly compact cardinals
The following question is motivated by a result of Magidor that it is consistent that the least strongly compact cardinal is the least measurable cardinal.
Question. Assume $\kappa$ is a strongly ...
10
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237
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Homogeneity of a variant of Prikry forcing
Prikry forcing is easily seen to be cone homogeneous (for any $p, q \in \mathbb{P}$, there are $p' \leq p, q' \leq q$ and an isomorphism $\Phi: \mathbb{P}/p' \simeq \mathbb{P}/q'$); in particular for ...
9
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Naive way to violate $\mathsf{SCH}$ at $\aleph_\omega$
I asked this question on MSE and got a partial answer. Shamefully I still haven't figured out myself a full answer, so I would like to ask it here. The usual way to get the failure of $\mathsf{SCH}$ ...
9
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260
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Can we have a 'universal class' for elementary embeddings $j\colon V\to V$
Work over $\mathsf{GB}$, Gödel-Bernays set theory (without choice). My question is the following:
Question. Is the following statement consistent with $\mathsf{GB}$? There is a universal class for ...
9
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535
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How much condensed mathematics can be founded on finite order arithmetic (or ETCS) instead of ZFC?
I recently learnt from David Roberts' answer that, there is a way, due to Colin McLarty, to set up the foundations on finite order arithmetic for EGA & SGA. In particular, all usages of ...
9
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279
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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 ...
9
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What are the known implications of “There exists a Berkeley cardinal”?
Inspired by this question: What are the known implications of "There exists a Reinhardt cardinal" in the theory "ZF + j"?
Definitions:
$\delta$ is Berkeley iff for every $\alpha\...
9
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249
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$\Sigma^2_2$ absoluteness and $\diamondsuit$
This question touches on recent questions concerning the role of $\diamondsuit$ and the Continuum Hypothesis. In particular, I would like to know more information about a conjecture of Woodin on the ...
9
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221
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Does there exist a non-trivial elementary embedding from an ultrapower $V^{I}/U$ to $V^{I}/U$?
Does there exist a set $I$ and an ultrafilter $U$ on $I$ and a non-trivial elementary embedding $j:V^{I}/U\rightarrow V^{I}/U$?
So the Kunen inconsistency result states that there does not exist a ...
9
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289
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ZF + "every Suslin set of reals is ${\bf \Sigma}^1_2$"
What is known about the theory
($\ast$) ZF + "every Suslin set of reals is ${\bf \Sigma}^1_2$"?
By "reals" I mean elements of the Baire space $\omega^\omega$. For a cardinal $\kappa$, a set of ...
9
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210
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What large cardinal axioms does the point of first difference between elementary embeddings satisfy?
Let $j,k:V_{\lambda}\rightarrow V_{\lambda}$ be inequivalent elementary embeddings. Then let $\theta(j,k)$ be the largest limit ordinal $\gamma$ such that $j(x)\cap V_{\gamma}=k(x)\cap V_{\gamma}$ for ...
9
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$\Pi_0^1$-weakly indescribable cardinals are exactly the regulars
Hi,
I'm not sure if I should ask here or over at math.stackexchange.com, but I think here it's a bit more fitting. This question stems from a homework problem:
Definition:
Given some class of formulas ...
8
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330
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Has there been any progress on this open problem about co-well-poweredness of accessible categories?
On the relations between accessible categories and large cardinal axioms, one big example is the following:
Assume the existence of a proper class of strongly compact cardinals. Then every accessible ...
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170
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What large cardinals are needed to imply projective sets have the perfect set property?
If there are infinitely many Woodins, then every projective set is determined, whence every projective set has the perfect set property (PSP). Since determinacy is such a stronger property than the ...
8
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217
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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 $\...
8
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411
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Choice function for elementary embedding $j: V_\lambda\prec V_\lambda$
Work in ZF. Let $\lambda$ be strongly inaccessible. Let $\kappa$ be such that for every $\alpha\lt\lambda$, there is some $j: V_\lambda\prec V_\lambda$, with critical point $\kappa$, such that $j(\...
8
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346
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Model for "$\kappa$ limit cardinal iff $2^\kappa$ limit cardinal"
Is there a model of ${\sf (ZFC)}$ such that in the model we have that $\kappa$ is a limit cardinal if and only if $2^\kappa$ is a limit cardinal?
8
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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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Is it true that $\kappa\to[\kappa]^2_2$ iff $\kappa\to[\kappa]^2_\kappa$ for inaccessible $\kappa$?
Recall that the square partition relation $\kappa\to[\lambda]^k_\eta$ holds iff for every $f:[\kappa]^k\to\eta$ there exists $H\in[\kappa]^\lambda$ such that $f"[H]^k\neq\eta$. I.e. said in words, ...
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Sharply less regular cardinals in set theory
If $\lambda<\kappa$ are two regular cardinals, then every locally $\lambda$-presentable category is locally $\kappa$-presentable. The analogous statement does not hold for $\lambda$-accessible ...
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Are there always large discrete families of normal measures?
Let $\kappa$ be a measurable cardinal. We give the Stone space of all ultrafilters on $\kappa$ the usual topology, where each $x\subseteq\kappa$ determines a basic open $[x]=\{U;x\in U\}$. The ...
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A question about strongly compact cardinals
Is the following equiconsistent with the existence of a strongly compact cardinals:
For every $\lambda > \kappa$ there exists a $\lambda$-strongly compact embedding $j: V \to M$ with the ...
8
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371
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PCF conjecture and fixed points of the $\aleph$-function
Recently Moti Gitik refuted Shelah's PCF conjecture, by producing a countable set $a$ of regular cardinals with $|\operatorname{pcf}(a)| \geq \aleph_1.$ See his papers
Short extenders forcings I and ...
8
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Idea behind the proof of consistency of club filter of $\omega_1$ is ultrafilter + ZF + DC
I've been trying to understand Radin Forcing and some of its applications, one of which is the use of it to prove the consistency of ''Club filter of $\omega_1$ is an ultrafilter + ZF + DC''. However, ...
7
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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 ...
7
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Is this determinacy principle consistent?
Let $\mathsf{ODet}_{\omega_1}(L(\mathbb{R}))$ be the following principle ("determinacy for simple open length-$\omega_1$ games"):
If $\kappa$ is any ordinal and $X\subseteq \kappa^{<\...
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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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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 ...
7
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$0^\#$ in weak theories vs large cardinals in $L$
To better understand the transition from large cardinal axioms consistent with the constructible universe $L$ to large cardinal axioms transcending $L$, I am looking for natural equiconsistencies ...
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If $j_{1},...,j_{n}:V_{\lambda+1}\rightarrow V_{\lambda+1}$ are elementary embeddings, then does $j_{1}(A)=...=j_{n}(A)=A$ for some linear order $A$?
Suppose that $j_{1},\dots,j_{n}:V_{\lambda+1}\rightarrow V_{\lambda+1}$ are elementary embeddings. Then does there necessarily exist a linear ordering $A$ of $V_{\lambda}$ such that $j_{1}(A)=\dots=j_{...