Questions tagged [large-cardinals]

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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 ...
tzimie's user avatar
  • 185
3 votes
0 answers
178 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 ...
Dmytro Taranovsky's user avatar
0 votes
0 answers
173 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_\...
Anindya's user avatar
  • 665
3 votes
0 answers
208 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 ...
Dmytro Taranovsky's user avatar
9 votes
2 answers
549 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 ...
Tim Campion's user avatar
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6 votes
0 answers
268 views

Some characterization of Mahlo cardinals

Following the well-known characterization of supercompact cardinals by Magidor, in our paper we have defined the notion of a $\kappa$-Magidor model, for supercompact cardinal $\kappa$. I defined a ...
Rahman. M's user avatar
  • 2,341
91 votes
10 answers
14k 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. ...
Peter Scholze's user avatar
4 votes
0 answers
173 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 ...
Asaf Karagila's user avatar
  • 38.1k
0 votes
0 answers
274 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 ...
Zuhair Al-Johar's user avatar
5 votes
0 answers
207 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,<...
Hanul Jeon's user avatar
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1 answer
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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 ...
ikrto's user avatar
  • 103
9 votes
0 answers
1k 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\...
Master's user avatar
  • 1,103
6 votes
0 answers
317 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 ...
Asaf Karagila's user avatar
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6 votes
1 answer
263 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 ...
Monroe Eskew's user avatar
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4 votes
1 answer
393 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 ...
Master's user avatar
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16 votes
1 answer
763 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 ...
Noah Schweber's user avatar
4 votes
0 answers
265 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_{\...
Master's user avatar
  • 1,103
5 votes
1 answer
170 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 ...
Hanul Jeon's user avatar
  • 2,774
8 votes
1 answer
314 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?
Someone211's user avatar
6 votes
0 answers
168 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....
Johannes Schürz's user avatar
4 votes
1 answer
582 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 $\lambda\lt\delta$, $...
Master's user avatar
  • 1,103
12 votes
1 answer
359 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 \...
Will Brian's user avatar
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6 votes
1 answer
361 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 ...
Noah Schweber's user avatar
3 votes
2 answers
1k 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 ...
Someone211's user avatar
2 votes
1 answer
527 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-...
Plasmath's user avatar
8 votes
0 answers
411 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(\...
Master's user avatar
  • 1,103
17 votes
2 answers
2k 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 ...
Monroe Eskew's user avatar
  • 18.1k
7 votes
1 answer
225 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 ...
Vincenzo Dimonte's user avatar
13 votes
1 answer
550 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 ...
Monroe Eskew's user avatar
  • 18.1k
34 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 ...
Jem's user avatar
  • 721
7 votes
1 answer
298 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 ...
Noah Schweber's user avatar
2 votes
0 answers
101 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 ...
Alec Rhea's user avatar
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6 votes
0 answers
202 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 ...
Jiachen Yuan's user avatar
17 votes
1 answer
749 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 $...
Noah Schweber's user avatar
1 vote
0 answers
224 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 ...
Alex O.'s user avatar
  • 51
4 votes
0 answers
432 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\...
Alex O.'s user avatar
  • 51
10 votes
1 answer
371 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 \...
Johannes Schürz's user avatar
5 votes
1 answer
263 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 ...
Monroe Eskew's user avatar
  • 18.1k
22 votes
1 answer
894 views

How badly can the GCH fail globally?

It's known that we can 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 ...
Sam Roberts's user avatar
  • 1,208
3 votes
1 answer
222 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 ...
Jesse Elliott's user avatar
6 votes
1 answer
225 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}(...
Monroe Eskew's user avatar
  • 18.1k
5 votes
0 answers
320 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 ...
Dmytro Taranovsky's user avatar
5 votes
0 answers
194 views

Can a weakly inaccessible non-weakly-Mahlo cardinal carry a $\kappa$-complete, $\kappa^+$-saturated ideal?

An ideal $I$ on a regular cardinal $\kappa$ is said to be $\mu$-saturated if whenever a family $\langle S_\alpha \mid \alpha<\lambda\rangle$ of subsets of $\kappa$ is such that each $S_\alpha\notin ...
Noah Schoem's user avatar
5 votes
0 answers
245 views

Absoluteness and the scale property for $Π^2_2$ or $Σ^2_2$

Under the diamond principle $◊$ and large cardinal axioms, which of the two pointclasses $Π^2_2$ or $Σ^2_2$ is expected to have the scale property? Because conditional $Σ^2_2$ absoluteness under $◊$ ...
Dmytro Taranovsky's user avatar
12 votes
1 answer
452 views

How to understand the interface of the consistency strength hierarchy, reverse mathematics, and proof-theoretic ordinal analysis?

I am aware of three major "hierarchies" of mathematical theories, but I don't know how to relate these hierarchies to one another. Here are the hierarchies I have in mind: Consistency strength. My ...
Tim Campion's user avatar
  • 60.6k
0 votes
0 answers
135 views

What's the consistency strength of resemblance + global failure of the continuum hypothesis?

Let $T$ be a theory formalized in first order logic with equality and membership and the additional primitive constant symbol $W$, with the following axioms: Extensionality: $\forall z (z \in x \...
Zuhair Al-Johar's user avatar
9 votes
0 answers
249 views

$\Sigma^2_2$ 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 ...
Todd Eisworth's user avatar
11 votes
0 answers
453 views

$\Sigma^2_1$ 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"...
Todd Eisworth's user avatar
14 votes
2 answers
415 views

Consequences of existence of a certain function from $\omega_1$ to $\omega_1$

In his book [1], Paul Larson remarks (Remark 1.1.22) that in L there is a function $h:\omega_1\rightarrow\omega_1$ such that for any countable elementary submodel $X$ of $V_\gamma$ (where $\gamma$ is ...
Todd Eisworth's user avatar
10 votes
1 answer
536 views

Is the product of commuting ultrafilters an ultrafilter?

If $U$ is a filter on $X$ and $W$ is a filter on $Y$, their product is the filter $U\times W$ on $X\times Y$ generated by rectangles $A\times B$ where $A\in U$ and $B\in W$. In certain circumstances ...
Gabe Goldberg's user avatar

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