The large-cardinals tag has no wiki summary.

**6**

votes

**1**answer

465 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 ...

**14**

votes

**3**answers

810 views

### Questions about $\aleph_1-$closed forcing notions

"Foreman`s maximality principle" states that every non-trivial forcing notion either adds a real or collapses some cardinals. This principle has many consequences including:
1) $GCH$ fails ...

**16**

votes

**1**answer

1k views

### Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)

Gödel's constructible universe ($L$) is defined using definable power set operator in first order logic ($\mathcal{L}_{\omega ,\omega}$). One can produce such a universe in infinitary logics in ...

**14**

votes

**3**answers

534 views

### Singularizing forcing of “small” cardinality?

Can there be a large cardinal $\kappa$ and a forcing of size $\kappa$ that makes $\kappa$ a singular cardinal? The motivation is that the standard Prikry forcing does not have a dense set of size ...

**4**

votes

**1**answer

246 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 = ...

**11**

votes

**1**answer

294 views

### cardinals below the critical point of a generic embedding

This may be an easy question. Are the cardinals below the critical point of a precipitous ideal embedding always absolute between the generic ultrapower and the generic extension?
To focus on the ...

**4**

votes

**1**answer

130 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

**1**answer

84 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 ...

**6**

votes

**1**answer

219 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 ...

**10**

votes

**2**answers

300 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 ...

**6**

votes

**0**answers

157 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

**1**answer

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 ...

**4**

votes

**2**answers

263 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

**0**answers

192 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

**0**answers

166 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 ...

**5**

votes

**0**answers

237 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

**3**answers

672 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

**0**answers

152 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

**1**answer

108 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

**2**answers

252 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

**1**answer

203 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

**1**answer

321 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

**1**answer

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

**2**answers

166 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 ...

**9**

votes

**0**answers

255 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 ...

**5**

votes

**2**answers

509 views

### Good Books about Large Cardinals

I am very new to set theory and have only learned the basics up to cardinal and ordinal aritmetic. I would like to learn about large cardinals and I am reading Thomas Jech's Set Theory. I have read ...

**4**

votes

**1**answer

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 ...

**4**

votes

**1**answer

153 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 ...

**7**

votes

**0**answers

170 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

**1**answer

222 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

**1**answer

200 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

**1**answer

210 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

**1**answer

217 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

**1**answer

230 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

**2**answers

967 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

**1**answer

215 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

**1**answer

284 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 ...

**6**

votes

**8**answers

2k views

### ULTRAINFINITISM, or a step beyond the transfinite

Cantor has, in the immortal words of D. Hilbert, given all of us a paradise (or perhaps, I would rather say, a great vacation spot), the TRANSFINITE.
$\aleph_0, \aleph_1,\aleph_2\dots$
the lists ...

**5**

votes

**0**answers

154 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 ...

**10**

votes

**2**answers

539 views

### Failure of diamond at large cardinals

What is known about the failure of $\Diamond_{\kappa}$ (diamond at $\kappa$) for $\kappa$ (the least) inaccessible, (the least) Mahlo and (the least) weakly compact.
Remark. The problem of forcing ...

**16**

votes

**4**answers

1k views

### What are the known implications of “There exists a Reinhardt cardinal” in the theory “ZF + j”?

This is, alas, in large part a series of questions on unpublished work of Hugh Woodin; it's also quite frivolous if Reinhardt cardinals turn out inconsistent.
Definitions:
Call $\kappa$ an ...

**10**

votes

**1**answer

362 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 ...

**10**

votes

**2**answers

619 views

### Questions about Prikry forcing and Cohen forcing

I have some questions.
The first one is about the product of Prikry's forcing.
Let $\kappa$ be a measurable cardinal, $U_1, U_2$ be normal measures on $\kappa$ and let $\mathbb{P}_{U_1}, ...

**12**

votes

**0**answers

363 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

**1**answer

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 ...

**8**

votes

**2**answers

251 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

**1**answer

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 ...

**6**

votes

**1**answer

224 views

### Solovay's Theorem on Partitions of Stationary Sets and Weak Choice Principles

There is a weak choice principle called $DC_\lambda$ which holds in $L(V_{\lambda+1})$ under the assumption of a non-trivial elementary embedding $$j:L(V_{\lambda+1})\prec L(V_{\lambda+1})$$ and it is ...

**5**

votes

**1**answer

150 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

**1**answer

159 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 ...