# Questions tagged [large-cardinals]

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604
questions

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votes

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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 ...

**5**

votes

**0**answers

144 views

### 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(...

**11**

votes

**1**answer

288 views

### 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 $V=L[r]$. ...

**4**

votes

**1**answer

239 views

### 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 ...

**7**

votes

**1**answer

244 views

### 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 ...

**6**

votes

**2**answers

311 views

### 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 ...

**7**

votes

**0**answers

174 views

### 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 ...

**3**

votes

**1**answer

289 views

### 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 ...

**3**

votes

**0**answers

113 views

### 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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votes

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130 views

### 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_\...

**2**

votes

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159 views

### 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 ...

**9**

votes

**2**answers

320 views

### 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 ...

**5**

votes

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158 views

### Some charactrization of Mahlo cardinals

Following the well-known characterization of supercpmpact cardinals by Magidor, in our paper we have defined the notion of a $\kappa$-Magidor model, for supercompact cardinal $\kappa$.
I defined a ...

**73**

votes

**10**answers

8k views

### Reflection principle vs universes

In category-theoretic discussions, there is often the temptation to look at the category of all abelian groups, or of all categories, etc., which quickly leads to the usual set-theoretic problems. ...

**4**

votes

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162 views

### Sequences of sequences of sequences and elementary embeddings

Suppose that $\kappa$ is the critical point of $j\colon V\to M$, and suppose that $\mathcal F=(F_\alpha\mid\alpha\leq\kappa)$ is a sequence such that for every limit ordinal $\alpha$, $F_\alpha$ is a ...

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votes

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221 views

### Can this Ackermann like set theory formulated without adding a primitive of set-hood reach the consistency of ORD is Mahlo?

The following is a theory that uses a reflection principle similar to Ackermann but on a size notion that is definable in the language of set theory, my question is about its consistency strength ...

**5**

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152 views

### Which very large cardinals are preserved under Woodin's forcing for $\mathsf{AC}$?

Woodin showed that we can force $\mathsf{AC}$ if there is a proper class of supercompact cardinals while preserving supercompacts, by forcing Easton-support iteration of $\operatorname{Col}(\kappa,<...

**0**

votes

**1**answer

139 views

### Coding a function $g:\kappa\to V_{\zeta+1}$ by an element of $V_{\zeta+1}$

Note: this is cross-posted from MSE.
This question is about the following remark (modified to be self-contained), found in Donald Martin's book on determinacy, page 340. The context is proving ...

**8**

votes

**0**answers

589 views

### 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\...

**6**

votes

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297 views

### Temporary destruction of measures in intermediate models

It is a well-known theorem that if $\kappa$ is measurable, then there is a generic extension in which $\kappa$ is no longer weakly compact, but we can force its weak compactness back and recover the ...

**6**

votes

**1**answer

218 views

### Tree property at weak inaccessibles

Suppose $\kappa$ is a weakly inaccessible cardinal with the tree property. What can we say about the height of $\kappa$? Is it a weakly-hyper-Mahlo of some sort? Does it enjoy some kind of ...

**4**

votes

**1**answer

350 views

### Does $H\vDash AC$

The set $H_\kappa$ of sets hereditarily of cardinality less than $\kappa$ is defined as $H_\kappa=\{x||tc(x)|\lt\kappa\}$. What if we define the set $H=H_{Ord}$ of sets hereditarily of cardinality ...

**15**

votes

**1**answer

514 views

### Can $Ord$ have nontrivial second-order elementary self-embeddings?

I forgot to mention originally: this was motivated by this old MSE question.
It's not hard to show in $\mathsf{ZFC}$ that there are nontrivial elementary embeddings from $(Ord; \in)$ to itself - or ...

**4**

votes

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195 views

### Reflection principles justifying $I2$ and larger cardinals

Consider the language of set theory together with the constant smybols $\langle \Omega_\alpha|\alpha\in Ord\rangle$. Let's add to $ZFC$ the axiom that for every $\alpha$,$n$ there is some $j: V_{\...

**5**

votes

**1**answer

142 views

### Finding many subsets of $V_{\lambda+2}$ stable under $j:V\prec V$

Working over $\mathsf{ZF}$ with an embedding $j:V\prec V$ with a critical point $\kappa$. Take $\lambda=\sup_{n<\omega}j^n(\kappa)$. (You may assume $\mathsf{DC}_\lambda$ if you need, but I am not ...

**8**

votes

**1**answer

226 views

### Inner model theory without choice

How much of the inner model project can be constructed without assuming the axiom of choice? I.e. which large cardinals provably have canonical inner models not assuming choice?

**6**

votes

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136 views

### Preserving supercompactness in intermediate forcing extensions

Let $\kappa$ be supercompact in $V$. Let $\mathbb{P}$ be one of the standard forcing notions (or an iteration of such), and for simplicity assume that $\mathbb{P}$ is ${<}\kappa$-directed closed (e....

**4**

votes

**1**answer

477 views

### What is the consistency strength of almost $\omega$-huge cardinals?

What is the consistency strength of a cardinal $\kappa$, such that there is some $j: V\prec M$ such that $M^{\lt j^\omega(\kappa)}\subseteq M$; in other words, for every cardinal $\lambda\lt\delta$, $...

**12**

votes

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248 views

### Getting a model of $\mathsf{ZFC}$ that fails to nicely cover an inner model

Consider the following statement:
$(\dagger)$ $\ $ There is an inner model $M$ such that $M \models \mathsf{GCH}+\square$ and for every countable $X \subseteq \mathrm{Ord}$, there is a countable $Y \...

**6**

votes

**1**answer

285 views

### How strong is “all up-classes are infinitarily definable”?

Working in MK (or some other not-too-strong class theory if you prefer), say that an up-class is a class of structures $\mathfrak{X}$ which is definable in $V$ (allowing parameters from $V$) and such ...

**3**

votes

**2**answers

950 views

### Inconsistency of Reinhardt cardinals in ZF+DC

As I'm just a layperson I don't understand the technicalities involved, but does the paper New Large Cardinal Axioms and the Ultimate-L Program, by Rupert McCallum (arXiv:1812.03837) prove the ...

**2**

votes

**1**answer

301 views

### Limit of Mahlo cardinals

What cardinal is the limit of this fundamental sequence?
{The first Mahlo cardinal, the first 1-Mahlo cardinal, the first hyper-Mahlo cardinal, the first hyper-hyper-Mahlo cardinal, the first hyper-...

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votes

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389 views

### 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(\...

**14**

votes

**1**answer

753 views

### A “paradox” about the inner model problem

As stated in Woodin, Davis, and Rodriguez - The HOD dichotomy, a longstanding open problem in set theory is to construct a canonical inner model for supercompactness. In general there are various ...

**7**

votes

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190 views

### Weakly homogenously Souslin sets and the measurability of $\omega_1$

I found this intriguing remark at the end of Woodin's Supercompact cardinals, sets of reals, and weakly homogeneous trees (1988):
The assertion that every set of reals, in $L(\mathbb{R})$, is the ...

**9**

votes

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229 views

### Iterating Neeman's forcing

In the paper, "Forcing with sequences of models of two types," (MR3201836), Neeman claims that, using a supercompact and a weakly compact above, one can force with his pure side conditions ...

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votes

**1**answer

3k views

### Is the theory Flow actually consistent?

Recently the paper
Adonai S. Sant'Anna, Otavio Bueno, Marcio P. P. de França, Renato Brodzinski, Flow: the Axiom of Choice is independent from the Partition Principle, arXiv:2010.03664
appeared on ...

**7**

votes

**1**answer

229 views

### Lowenheim-Skolem numbers for SOL + correctness quantifiers

For a logic $\mathcal{L}$, say that a cardinal $\kappa$ is $\mathcal{L}$-correct iff every satisfiable $\mathcal{L}$-theory of size $<\kappa$ has a model of size $<\kappa$. First-order ...

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votes

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80 views

### Consistency strength of iterated classes

Adding classes into a set theory like ${\bf ZFC}$ to get a theory like ${\bf MK}$ adds some consistency strength, but less than even a single inaccessible cardinal since $\kappa$ being inaccessible ...

**6**

votes

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172 views

### Collapse successor of singular while preseving supercompactness

Suppose $\kappa$ is a supercompact cardinal. Is it possible to find a forcing which collapses $\kappa^{+\omega+1}$ to $\kappa^{+\omega}$ (all those $\kappa^{+n}$'s are preserved) while the ...

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304 views

### What sets can be unraveled?

A set $X\subseteq\omega^\omega$ is unravelable iff there is a possibly larger set $A$ and a clopen set $Y\subseteq A^\omega$ (with respect to the product topology coming from the discrete topology on $...

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vote

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176 views

### Weakly berkeley cardinal

Define $\kappa$ as $\Sigma_n$-weakly berkeley cardinal if for any transitive set $M$ that includes $\kappa$ exist elementary embedding $j:M\rightarrow M$ save only $\Sigma_n$ formulas and critical ...

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votes

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248 views

### Are hyper-Berkeley cardinals equiconsistent with club Berkeley cardinals or with Berkeley cardinals?

Let's define cardinal $\kappa$ as hyper-Berkeley if for any transitive set $M$ such that $\kappa\in M$ there exists an elementary embedding $j: M\prec M$ with
fixed point $\lambda$ and $\text{crit}j\...

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votes

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324 views

### The stationary reaping number $\mathfrak{r}_{cl}$

Let $\kappa$ be at least inaccessible (but measurable is what I am primarily interested at the moment). Let $x,y \in [\kappa]^\kappa$ both be stationary. We say that $y$ stationary-splits $x$ iff $x \...

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votes

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245 views

### Consistency strength of lifting through a lot of collapsing

What is the consistency strength of the following situation?
$j : V \to M$ is an elementary embedding definable from parameters in $V$, with critical point $\kappa$.
$\mathbb P$ is a forcing that ...

**19**

votes

**1**answer

650 views

### How badly can the GCH fail globally?

It's known that we have global failures of GCH---for example, where $\forall \lambda(2^\lambda = \lambda^{++})$---given suitable large cardinal axioms.
My question is whether we can have global ...

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votes

**1**answer

192 views

### smallest ordinal $\alpha$ such that $L \cap P(L_\alpha)$ is uncountable

Let $V$ denote the von Neumann universe and $L$ Gödel's constructible universe. For any set $X$, let $P(X)$ denote the power set of $X$.
Assume that $0^\sharp$ exists (and ZFC).
What is the smallest ...

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votes

**1**answer

206 views

### Can we recover an inner model of CH after forgetting some generic information?

Suppose $\kappa$ is an inaccessible cardinal. Let $G \times H$ be $\mathrm{Col}(\omega_1,{<}\kappa) \times \mathrm{Add}(\omega,\kappa)$-generic over $V$. Let $X \subseteq \kappa$ be $\mathrm{Add}(...

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79 views

### At which large cardinal property Ackermann set theory + finitization rule would stop?

By the finitization rule I mean a rule that inputs a schema in the $V$ world and outputs a single statement in the $V$ world that serves to capture that schema!
So in this sense we'll have for every ...

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215 views

### Inner models with all sets generic

Question: Under large cardinal axioms, what is the intersection of all inner models $M$ of ZFC such that every set in $V$ is set-generic over $M$?
Every set belongs to a generic extension of HOD, and ...