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5
votes
2answers
113 views

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 ...
4
votes
1answer
182 views

$\text{ZFGC}^{\text{−f}}+\text{BAFA}+\exists\kappa(κ \text{ is Reinhardt})$ 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 ...
4
votes
1answer
303 views

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 ...
17
votes
1answer
1k views

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 ...
4
votes
1answer
132 views

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 ...
3
votes
1answer
86 views

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 ...
4
votes
1answer
257 views

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 = ...
6
votes
1answer
224 views

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 ...
6
votes
0answers
160 views

Singular Jonsson cardinals

Is the consistency of the following well-known: $(*)$: There exists a singular cardinal $\kappa$ such that : (1) $\kappa$ is a Jonsson cardinal, (2) $\kappa$ is not a fixed point of the ...
5
votes
1answer
141 views

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 ...
10
votes
2answers
315 views

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 ...
4
votes
2answers
268 views

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 ...
6
votes
0answers
167 views

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 ...
6
votes
0answers
199 views

$\delta$-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 ...
6
votes
1answer
479 views

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 ...
5
votes
0answers
241 views

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 ...
12
votes
3answers
674 views

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 ...
9
votes
0answers
153 views

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. ...
3
votes
1answer
109 views

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 $T$ ...
12
votes
2answers
254 views

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 ...
9
votes
1answer
209 views

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 ...
11
votes
1answer
325 views

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 ...
4
votes
1answer
147 views

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 ...
8
votes
2answers
167 views

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 ...
4
votes
1answer
157 views

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 ...
9
votes
0answers
256 views

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 ...
7
votes
0answers
173 views

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} ...
6
votes
1answer
226 views

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 ...
3
votes
1answer
203 views

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 ...
8
votes
1answer
213 views

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 ...
4
votes
1answer
223 views

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 ...
10
votes
1answer
232 views

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 ...
8
votes
2answers
986 views

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 ...
6
votes
1answer
218 views

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 ...
6
votes
1answer
290 views

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 ...
10
votes
1answer
365 views

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 ...
5
votes
0answers
156 views

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 ...
12
votes
0answers
368 views

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 ...
7
votes
1answer
250 views

Inaccessible becomes successor of singular

Is it possible, starting from any large cardinal assumption, to find a countably closed forcing $\mathbb{P}$ such that for some inaccessible $\kappa$, $\Vdash_\mathbb{P} "\kappa = \lambda^+$ and ...
4
votes
1answer
201 views

The word problem of the free left distributive algebra on one generator

A left distributive algebra is a set $A$ together with a binary operation, $\cdot$, satisfying $a\cdot(b\cdot c)=(a\cdot b)\cdot(a\cdot c)$. One important example of left distributive algebras arises ...
8
votes
2answers
254 views

A Weakening of the Tree Property

If $f$ and $g$ are two functions, define $f \sim g$ if they differ only finitely often on their common domain. The following property of a large cardinal arose from a problem in model theory. I am ...
8
votes
1answer
127 views

Strongly compact cardinal with bad covering properties

This is a continuation of the question covering properties of strongly compact embedding. Recall that a cardinal $\kappa$ is $\nu$-strongly compact cardinal if there is an elementary embedding ...
5
votes
1answer
151 views

A realcompact analogue of the Baire category theorem

Let $\frak{m}$ be the least measurable cardinal. A space $X$ is realcompact if it is homeomorphic to a closed subset of some product $\mathbb{R}^I$. Let $X$ be realcompact with $P_\frak{m}$ topology, ...
5
votes
1answer
161 views

Covering properties of strongly compact embedding

Let $\kappa$ be a $\mu$-strongly compact cardinal, which means that there is an elementary embedding $j:V\rightarrow M$, with critical point $\kappa$ such that $M$ is well founded (even closed under ...
11
votes
1answer
314 views

What is the large cardinal strength of the assertion that every $\kappa$-complete filter on $\kappa$ extends to a $\kappa$-complete ultrafilter?

It is well-known that an uncountable regular cardinal $\kappa$ is strongly compact if and only if every $\kappa$-complete filter on any set extends to a $\kappa$-complete ultrafilter on that set. The ...
6
votes
0answers
203 views

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," ...
10
votes
1answer
458 views

Does Turing determinacy imply full determinacy?

The axiom of Turing determinacy is a weakening of the full axiom of determinacy, $AD$, in which only games with payoff sets which are $\equiv_T$-invariant are demanded to be determined. In "Turing ...
3
votes
2answers
450 views

A question about “local” versus “global” large cardinal axioms

The terms "local" and "global" when applied to large cardinal axioms seem to have a well understood intuitive meaning, although a formalized definition of them in (a meta-language for)ZFC might be ...
3
votes
1answer
161 views

Are normal ultrafilters generated by conditional closure systems?

Suppose that $\kappa$ is a cardinal, $X$ is a set with $|X|>\kappa$, and $\mathcal{U}\subseteq P(P_{\kappa}(X))$ is a normal ultrafilter. We say that a collection $C\subseteq P_{\kappa}(X)$ is a ...
5
votes
1answer
210 views

Is there a large-cardinal completeness theorem for $L$?

I cannot currently find the original, but if memory serves, Goedel once speculated that there might be a "large-cardinal completeness theorem for $V.$" This theorem would state: *Theorem. For every ...