Questions tagged [large-cardinals]
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773
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Around Vopěnka: Accessible category with small full discrete subcategories of arbitrary size?
I believe the model-theoretic version of the question is: is there a theory in finitary first-order logic which has, for each cardinal $\lambda$, a set $C_\lambda$ of $\lambda$-many models, such that ...
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2
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797
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Whatever happened to $L(j)$?
So this question probably shows my inner model theoretic ignorance, but:
In "Two remarks on elementary embeddings of the universe" (http://projecteuclid.org/download/pdf_1/euclid.pjm/1102969567), ...
- 19.1k
5
votes
1
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Universally Baire Tree Representation of Projective Sets
In Feng, Magidor, and Woodin "Universally Baire Sets of Reals", they show that if $A$ is a $\mathbf{\Pi}_2^1$ set and $U$ and $V$ are any pair of trees witnessing the universal baireness of $A$, then ...
- 1,830
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1
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Are the failure of SCH and "$cf([\mu]^{cf (\mu)},\subset)>\mu^+$ for some singular" equiconsistent?
Is it true that the following two statements are equiconsistent?
(1) $2^\mu>\mu^+$ for some strong limit singular cardinal $\mu$
(2) $cf([\mu]^{cf (\mu)},\subset)>\mu^+$ for some singular ...
- 1,477
4
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1
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409
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Name for an intermediate notion between huge and 2-huge
I am employing a large cardinal notion that has been used explicitly before, and I am wondering if someone has given it a good succinct name.
A cardinal $\kappa$ is huge if there is an elementary $j : ...
- 18.1k
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2
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1k
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When does Vopěnka's principle hold?
Vopěnka's principle (VP) is the statement that, given any proper class $\{\mathcal{A}_\eta: \eta\in ON\}$ of first-order structures in the same language, there are some $\alpha\not=\beta$ with $\...
- 19.1k
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1
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Explicit counter example to Vopěnka's principle in the constructible universe?
Vopěnka's principle is a large cardinal axiom which has many equivalent formulations. One of them, which I find especially appealing, is the following: if the universe is satisfies Vopěnka's principle ...
- 27.2k
6
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2
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360
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The role of the rigid relation principle ($RR$) in the Kunen inconsistency
Consider the rigid relation ($RR$) principle, i.e.
"every set admits a rigid binary relation", that is,"that for every set $A$ there is a binary relation $R$ on $A$ such that the structure $(A,R)$ is ...
- 6,024
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1
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270
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Existence of $\kappa$-Suslin trees above a measurable cardinal
We have learned from Joel David Hamkins and Monroe Eskew that:
Answers:
Having a measurable cardinal $\delta$ we can force a $\kappa$-Suslin tree for many $\kappa$'s above $\delta$.
But is the ...
- 3,031
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Consistency strength of $\aleph_2$-Souslin hypothesis
Question 1. What is known about the consistency strength of $\aleph_2$-Souslin hypothesis?
Question 2. What is known about the consistency strength of having both $\aleph_2$-Souslin hypotheis and $\...
9
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Can the first ordinal in which $V\neq HOD$ be $\aleph_\omega$?
Assume that $V\neq HOD$ and let $\kappa = \min \{\alpha\in On \mid \mathcal{P}(\alpha) \not\subseteq HOD\}$.
Clearly, $\kappa$ is a cardinal.
Question: Is it consistent that $\kappa = \aleph_\...
- 5,192
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2
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325
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Reference for proof that consistency of $\omega_1$-Erdos cardinal implies Con(Chang's Conjecture)
What is a good source for Silver's proof (or a more modern version) that Con($\exists \omega_1$-Erdos cardinal) implies Con(Chang's Conjecture)?
Silver's original proof seems to have never been ...
- 2,209
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1
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$\text{ZFGC}^{\text{−f}}+\text{BAFA}+\exists\kappa(κ \text{ is Reinhardt})$ and its implication
A. S. Daghighi, M. Golshani, J. D. Hamkins, and E. Jeřábek proved in "The foundation axiom and elementary self-embeddings of the universe" that, working in ZFGC$^{\text{−f}}$+BAFA, there are ...
10
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If $\kappa$ is weakly inaccessible and $A\subset\kappa$, can $L[A]$ violate $\kappa^{\lt\kappa}=\kappa$?
In some current work, my co-authors and I had wanted in a certain
argument to appeal to $\kappa^{\lt\kappa}=\kappa$ in $L[A]$, in a
situation where $A\subset\kappa$ and $\kappa$ was weakly
...
- 225k
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Inaccessible cardinal and $\Sigma_1$ reflection
A theorem of A. Levy says that, if $\kappa$ is an inaccessible cardinal, then $V_\kappa\prec_{\Sigma_1}V$ namely $V_\kappa$ is an elementary submodel when considering only $\Sigma_1$ formulas.
Where ...
- 115
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Philosophical arguments in defense (or against) large cardinals
The question is essentially what is asked in the title. I split it into two parts
(A) (Arguments supporting the existence of large cardinals) What are the main philosophical arguments in defense of ...
4
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Presaturated ideals
In this paper, Gitik and Shelah make the following claim (part of Proposition 1.5):
Claim (Gitik-Shelah): Suppose $\kappa < \lambda$ are regular, $2^\lambda = \lambda^+$, and $D$ is a normal ...
- 18.1k
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1
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The GCH in a reverse Easton support iteration
I am trying to understand the proof that the GCH can first fail at a weakly compact cardinal. We assume the GCH and that there exists a weakly compact cardinal $\kappa$, and we construct a reverse ...
- 1,688
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1
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stationary tower forcing
It is known that if $\delta$ is a Woodin cardinal and $\kappa < \delta$, then the stationary tower forcing $\mathbb Q^\kappa_{<\delta}$ preserves cardinals up to $\kappa$ and forces $\delta = \...
- 18.1k
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Is there a simple combinatorial characterization for when a direct limit of ultrapowers of $V$ is well-founded?
I want to know if there are fairly simple combinatorial necessary conditions for when a direct limit of ultrapowers of $V$ is well-founded similar to $\sigma$-completeness. By combinatorial, I mean ...
- 27.8k
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364
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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 ...
7
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1
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Pseudo-Prikry sequences vs Prikry sequences
Definition: Let $V\subseteq W$ be two transitive models of $ZFC$. A pseudo-Prikry sequence, $s$, at a cardinal $\kappa$ for $(V, W)$ is an $\omega$-sequence, cofinal at $\kappa$ such that for every ...
- 5,192
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3
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Is there a compendium of the consistency strength between the most important formal theories?
Preliminar Notions:
A formal system is a tuple $(\Sigma,G,A,R)$ where $\Sigma$ is an alphabet (set of symbols), $G$ is a formal grammar on $\Sigma$ that generates a formal language $L$ (set of well ...
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515
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Limits of determinacy on reals
For $X\subseteq\mathbb{R}^\omega$, say that $X$ is determined if the associated game on $\mathbb{R}$ of length $\omega$ (players I and II alternate playing reals, player I wins iff the sequence built ...
- 19.1k
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0
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289
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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 ...
7
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429
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$\delta$-strong compactness and generalized strong tree properties
Are there non-trivial equivalent characterizations of $\delta$-strongly compact (and almost strongly compact) cardinals in terms of generalized tree properties?
Recall the definitions as per Joan ...
- 2,121
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756
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A question on rank-to-rank embeddings
Consider a non-trivial elementary embedding $j:V_\lambda\to V_\lambda$ and, for each $A\subset V_\lambda$, set $j(A)=\bigcup_{\delta<\lambda}j(A\cap V_\delta)$.
In Implications between strong ...
- 1,688
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0
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419
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Cardinal characteristics without choice
(I'm taking my definition of a cardinal characteristic from Blass' excellent article http://www.math.lsa.umich.edu/~ablass/need.pdf, which cites Vojtas/Fremlin/Miller; theirs is more general, but I'm ...
- 19.1k
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3
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Complete resolutions of GCH
Let's say that a "complete resolution of GCH" is a definable class function $F: \operatorname{Ord}\longrightarrow \operatorname{ Ord}$ such that $2^{\aleph_\alpha} = \aleph_{F(\alpha)}$ for all ...
- 4,123
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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. ...
- 2,121
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Classify set theories whose transitive models sharing the same sets of ordinals are equal
This question is a follow-up from my recent question, Classifying set theories whose standard models sharing the same ordinals are equal
Let's say that a (recursively axiomatizable) set theory $T$ ...
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800
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Classifying set theories whose standard models sharing the same ordinals are equal
Let's say that a (recursively axiomatizable) set theory $T$ extending ZF is "ordinal-categorical" if, whenever $M$ and $N$ are standard models of $T$ sharing the same ordinals, one has $M = N$. For ...
- 4,123
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970
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Strongest large cardinal axiom compatible with $V = L$?
What is the strongest known natural large cardinal axiom compatible with $V = L$ (strongest in the sense that it implies all known "small" large cardinal axioms, where a large cardinal axiom is said ...
- 4,123
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1
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What are the current views on consistency of Reinhardt cardinals without AC?
It's well known that Reinhardt cardinals are inconsistent, provided that we have access to axiom of choice, but, as far as I know, we are clueless about this when we don't assume choice. For me, the ...
5
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1
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471
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Simplest non-constructible set of integers compatible with the nonexistence of $0^\sharp$?
What is the simplest non-constructible set of integers (say, in the analytical hierarchy) that is compatible with the nonexistence of $0^\sharp$? In particular, can there still be a non-constructible $...
- 4,123
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2
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384
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Elementary embeddings with the same critical point
Question: Is it consistent (relative to the existence of large cardinals) that there is an elementary embedding $j\colon V\to M$ (where $M$ is transitive model) that factors as $j = j_n \circ k_n$ for ...
- 5,192
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1
answer
237
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almost huge embeddings and stationary correctness
Suppose $\kappa$ is an almost huge cardinal. Using the characterization of Theorem 24.11 in Kanamori's book, one can show that if $j : V \to M$ is an embedding derived from an almost-huge tower of ...
- 18.1k
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Reinhardt cardinals and iterability
Work in $ZF$. Let $j:V\to V$ be a non-trivial elementary embedding which is iterable, so that we can iterate it and form models $M_\alpha, \alpha\in ON,$ with $M_0=V,$ and elementary embeddings $j_{\...
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0
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237
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Countable choice in $L(\mathbb{R}^*_G)$
Let $\lambda$ be a singular strong limit cardinal and let $G \subset \text{Col}(\omega,\mathord{<}\lambda)$ be a $V$-generic filter. Let $\mathbb{R}^*_G = \bigcup_{\alpha < \lambda} \mathbb{R}^{...
- 5,551
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Can the Cohen forcing collapse cardinals?
Let $\kappa$ be a regular cardinal, and let $\mathbb{P} = Add(\kappa,1)$ be the standard forcing notion for adding a new subset of $\kappa$ using partial function from $\kappa$ to $2$ with domain of ...
- 5,192
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1
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How to change the successor of a singular with a Woodin?
I'm looking for references on how to change the successor of a singular cardinal from "more or less" minimal assumptions. If possible, then without adding bounded subsets to the singular either.
In ...
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1
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426
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Erdős cardinals and $0^\sharp$
It is well-known that if the Erdős cardinal $\kappa(\omega_1)$ exists, then $0^\sharp$ exists, but what if $\kappa(\lambda)$ exists for a limit ordinal $\omega_1^L\leq \lambda<\omega_1$? Does this ...
- 1,688
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1
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504
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Iterated ultrapowers of L
If there exists a measurable cardinal, we can generate a sequence of iterated ultrapowers $\{Ult_U^\alpha(V)\}_{\alpha\in ON}$. If $0^\sharp$ exists, i.e. if there exists an elementary embedding $j:L\...
- 1,688
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334
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On the definition of the $\alpha$-iterable cardinals
I am reading the paper Ramsey-like cardinals II by Victoria Gitman and Philip Welch (Journal of Symbolic Logic, vol. 76, no. 2. pp. 541-560, 2011) and maybe I am missing something.
According to the ...
- 1,688
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The impact of large cardinals in mathematics [closed]
What are the main applications of large cardinals in ordinary mathematics, and what is the philosophy behind using them. In particular:
Question 1. What is the philosophy behind accepting large ...
8
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1
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404
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On a weak tree property for inaccessible cardinals
Suppose that $\kappa$ is inaccessible and consider a tree of height $\kappa$ whose levels have size strictly below some cardinal $\gamma < \kappa$. Does this type of tree always have a $\kappa$-...
- 5,692
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1
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567
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higher-order reflection
In the first-order context, "reflection" of a formula $\varphi(x)$ below $\kappa$ refers to the the following situation:
There are many ordinals $\alpha<\kappa$ such that for all $a \in V_\...
- 18.1k
11
votes
1
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810
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Erdős cardinals and ineffable cardinals
In Cantor's Attic it is stated that an $\omega$-Erdős cardinal is a stationary limit of ineffable cardinals, and Jech book is given as a reference, but I cannot find this result in that book. I have ...
- 1,688
5
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0
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224
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A variant of Chang's model with choice
Let $M_n$, $n < \omega$, be a models of $ZFC$ with the same ordinals, closed under countable sequences. Let $\alpha_n$ be an ordinal which is a regular cardinal in $M_n$.
Question 1: Is it ...
- 5,192
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757
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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 ...