Questions tagged [large-cardinals]

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Can we interpret Reinhardt cardinals this way?

To the language of set theory add a primitive unary predicate $\operatorname {Universe}$ and a primitive total unary function $j$. Add all axioms of $\sf ZF$ in the language of this theory, i.e. the ...
Zuhair Al-Johar's user avatar
9 votes
0 answers
260 views

Can we have a 'universal class' for elementary embeddings $j\colon V\to V$

Work over $\mathsf{GB}$, Gödel-Bernays set theory (without choice). My question is the following: Question. Is the following statement consistent with $\mathsf{GB}$? There is a universal class for ...
Hanul Jeon's user avatar
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1 vote
2 answers
258 views

Can we iteratively reflect on self elementary embeddable stages of the cumulative hierarchy?

Is it possible to iterate elementary embeddability and reflect on those stages that are elementary embeddable to themselves? The following is a formal capture of that idea: To the language of $\sf ZF$...
Zuhair Al-Johar's user avatar
8 votes
1 answer
1k views

Is there a form of choice that can elude Kunen's inconsistency theorem?

When it is said that Kunen inconsistency theorem proves that given $\sf ZFC$ no elementary embedding can exist from the universe to itself. Most references quote full choice in stating that result, ...
Zuhair Al-Johar's user avatar
3 votes
1 answer
268 views

If we add stratified\acyclic replacement to the wholeness axiom, would that increase its consistency strength?

If we add to the wholeness axiom, the axiom of stratified\acyclic replacement, what would be the consistency state and strength of the resulting theory? The wholeness axiom $\sf WA$, introduced by ...
Zuhair Al-Johar's user avatar
2 votes
0 answers
124 views

Cat as a bicategory of monads over another category

Let's assume infinitely many Grothendieck universes exist. Let's call $\kappa$-Cat the bicategory of $\kappa$-small categories with anafunctors and anatural transformations. Now for any $\lambda$ and ...
Ilk's user avatar
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3 votes
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If a theory speaks of sets that cannot be forced to be parameter free definable, then does this entail a large cardinal property?

If we say that an effectively generated first order theory $\sf T$ extends $\sf ZF$, such that every countable model of $\sf T$ doesn't have a class forcing extension that is pointwise definable. ...
Zuhair Al-Johar's user avatar
10 votes
0 answers
304 views

Feferman's universes for proof assistants?

This question was prompted by a discussion from another MO question about the consistency of ZFC. There are some mathematicians who are comfortable with ZFC but uneasy with large cardinals. For them,...
Timothy Chow's user avatar
0 votes
1 answer
206 views

Where do the universe embedds to in Reinhardt's cardinals setting?

I just want to understand the embedding behind Reinhardt's cardinals. We have an elementary embedding $j: V \to V$. Let the background theory be $\sf MK - Choice$. We know that $V$ itself is a class ...
Zuhair Al-Johar's user avatar
8 votes
1 answer
399 views

Compatibility of $\mathsf{SVC}$ and Reinhardtness

Can we prove the consistency of $\mathsf{ZF+SVC}$ + "There is a Reinhardt cardinal?" (Preferably from the consistency of $\mathsf{ZF}$ with a Reinhardt cardinal, but using a stronger ...
Hanul Jeon's user avatar
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"Very $L$-like" models, part 2: combinatorics

Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ and having the finite use property and the strong downward Lowenheim-Skolem property together with, for each finite ...
Noah Schweber's user avatar
8 votes
1 answer
241 views

Some relevant questions about the consistency strength of singularity of $\omega_1$ and $\omega_2$

The following question was asked years ago on MSE, but let me recap it: Question: Is there anything currently known about the exact consistency strength of "$\mathsf{ZF}$ + both $\omega_1$ and $\...
Hanul Jeon's user avatar
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8 votes
1 answer
311 views

First inaccessible Suslin trees in L, an interesting detail

It's known (but quite nontrivial) that $V=L$ implies that if $\kappa$ is the 1st inaccessible cardinal then there are $\kappa$-Suslin trees $T$. Such a tree $T$ can be considered as a forcing notion ...
Vladimir Kanovei's user avatar
9 votes
0 answers
535 views

How much condensed mathematics can be founded on finite order arithmetic (or ETCS) instead of ZFC?

I recently learnt from David Roberts' answer that, there is a way, due to Colin McLarty, to set up the foundations on finite order arithmetic for EGA & SGA. In particular, all usages of ...
Z. M's user avatar
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5 votes
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180 views

"Very $L$-like" models, part 1: large cardinals

(The original version of this question was much narrower and less natural; but see the edit history if interested.) Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ ...
Noah Schweber's user avatar
2 votes
1 answer
190 views

Large cardinals and measurability in $L(A)$

Under suitable large-cardinal assumptions, in the inner model $L(\mathbb R)$ one can have $\omega_1$ and $\omega_2$ measurable (this follows from determinacy). I was wondering if it is possible to ...
Curious_poster's user avatar
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0 answers
136 views

Weakening of open determinacy for uncountably long games

For a cardinal $\kappa$ I'll use the phrase "$(\kappa,\kappa)$-game" to mean "two-player, perfect-information, deterministic game on $\kappa$ of length $\kappa$." Say that a ...
Noah Schweber's user avatar
8 votes
0 answers
217 views

Absoluteness of the core model under a proper class of completely Jónsson cardinals

Example 2.4.2 of Larson's The stationary tower (based on Woodin's lecture) describes how we can absorb a generic filter over a set forcing in a definable inner model into a generic extension by $\...
Hanul Jeon's user avatar
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8 votes
1 answer
350 views

On the strength of higher-logic analogues of $\mathsf{ZFC}$ + Montague's Reflection Principle

Throughout, I work in $\mathsf{MK}$ in order to be able to conveniently quantify over logics; if one prefers, we can restrict attention to (say) $\Sigma_{17}$-definable logics and work in $\mathsf{ZFC}...
Noah Schweber's user avatar
5 votes
1 answer
197 views

Upwards-fragility of inaccessibles (again)

Although self-contained, this question is a follow-up to this earlier one. Also, thanks to Fedor Pakhomov for fixing a trivial early version of this question! Work in $\mathsf{ZFC}$ + "There is a ...
Noah Schweber's user avatar
4 votes
0 answers
114 views

Canonical functions on $\kappa$ and canonical stationary sets

This is a somewhat continuation of this question. The related paper is Jech's Stationary subsets of inaccessible cardinals. See also Chapter 8 and Chapter 24 of Jech's Set Theory. I would like to ask ...
Clement Yung's user avatar
5 votes
1 answer
172 views

Fragility of large cardinals with respect to transitive end extensions

To motivate things, let me start with a special case of the question I'm interested in. Let $\mathsf{In}(x)\equiv$ "$x$ is an inaccessible cardinal." Question 1: Is it consistent with the ...
Noah Schweber's user avatar
5 votes
0 answers
240 views

How strong is this "modal definability principle"?

Throughout, we work in the class theory $\mathsf{MK}$ (although I'm open to tweaking this), "logic" means "set-sized logic whose semantics is definable over $V$," and "$\...
Noah Schweber's user avatar
7 votes
2 answers
718 views

Why are stationary limits so ubiquitous when studying large cardinals?

While studying large cardinals, I have frequently noticed the following phenomenon: If X and Y are two different types of large cardinals, then every cardinal of type X is a stationary limit of ...
Anindya's user avatar
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12 votes
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Does $2^{\aleph_0}\rightarrow [\aleph_1]^2_3$ require that the continuum is weakly inaccessible?

A classic result of Sierpiński shows that $2^{\aleph_0}\nrightarrow [\aleph_1]^2_2$, that is, there is a coloring of pairs of real numbers using two colors such that both colors appear on any ...
Todd Eisworth's user avatar
7 votes
0 answers
166 views

Intuition for branch uniqueness in inner model theory

In inner model theory, what is the intuition behind the expectation that under appropriate conditions, we should have a single preferred branch to continue an iteration at a limit stage? At the level ...
Dmytro Taranovsky's user avatar
6 votes
1 answer
236 views

Can there be no complexity bound on the definable elementary $V\rightarrow M$?

This starts with a vaguely-recalled result (which may be false!): that if $\mathcal{U}$ is a measure on the least measurable cardinal $\kappa$, then every elementary $j: L[\mathcal{U}]\rightarrow M$ ...
Noah Schweber's user avatar
3 votes
0 answers
144 views

Inner model theory using indiscernibles

Has an inner model theory been developed on the basis of indiscernibles rather than measures? Is there a reasonable formalization at the level of overlapping extenders? Fine-structural models beyond $...
Dmytro Taranovsky's user avatar
3 votes
1 answer
600 views

How would one formulate large cardinals beyond rank into rank?

Crossposting from MSE, after deciding that this question is related to modern research in set theory: https://math.stackexchange.com/questions/4499391/how-would-one-formulate-large-cardinals-beyond-...
littleman's user avatar
  • 201
7 votes
1 answer
402 views

Are large cardinals about more than just consistency?

The other day, I was reading the preface of Kanamori's The Higher Infinite and noticed that he says large cardinals provide a useful 'measuring stick' for consistency. That raised the question of ...
littleman's user avatar
  • 201
1 vote
1 answer
122 views

Would loss of downward absoluteness for large cardinals repeat itself upwardly?

if $\kappa$ is a cardinal such that $V_\kappa \models \sf ZFC$, then $\kappa$ is called a worldly cardinal and this is not necessarily downward absolute [Hamkins], i.e. there is a model $V$ of $\sf ...
Zuhair Al-Johar's user avatar
3 votes
0 answers
119 views

Greatly Erdős and Erdős cardinals

Sharpe and Welch 2011 define $\alpha$-weakly Erdős and greatly Erdős cardinals as follows: Let $\kappa$ have uncountable cofinality, and let $\mathcal{A}$ be a $\kappa$-structure, $X \subseteq \kappa$...
Arvid Samuelsson's user avatar
5 votes
0 answers
143 views

Ramsey-theoretic properties of Erdős cardinals

The $\beta$-Erdős cardinal is defined as the least ordinal $\eta$ such that $\eta \to (\beta)^{\lt \omega}_2$, that is for every function $f: [\eta]^{\lt \omega} \to 2$, there is an $f$-homogeneous ...
Arvid Samuelsson's user avatar
1 vote
0 answers
103 views

What's the consistency strength of adding this inference rule to Ackermann's set theory?

Working in the language of Ackermann set theory: Let $\phi(\vec{P},\vec{x})$ be a formula not using the symbol $V$, where $\vec{P}$ are predicate symbols definable in the language of set theory, and $\...
Zuhair Al-Johar's user avatar
3 votes
0 answers
120 views

At which large cardinal property this second order ordinal arithmetic stops?

Language: Second order logic, with as usual predicates written in upper case, and objects in lower case. Let $<$ be a primitive constant binary relation symbol. Equality between objects is ...
Zuhair Al-Johar's user avatar
2 votes
0 answers
109 views

Consistency strength of Muller's modification of Ackermann set theory

In his 2001 paper Sets, Classes and Categories, F. A. Muller lays out a modification of Ackermann class theory that he claims is not conservative over Ackermann's original theory in the sense that it ...
Alec Rhea's user avatar
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1 vote
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143 views

What's the consistency strength of this kind of cardinal?

Bumped, since I was recently thinking about these again. My friend introduced the following notion: Let $\xi > 0$, $\eta$ be ordinals, $n$ be a natural number and $\mathcal{A}, X$ be classes. A ...
Binary198's user avatar
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2 votes
1 answer
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Is this strengthening of the definition of weakly Shelah cardinals equivalent to being weakly Shelah?

The MathOverflow question A weak (?) form of Shelah cardinals by Trevor Wilson defines weakly Shelah cardinals as follows: A cardinal $\kappa$ is weakly Shelah if for all $f : \kappa \to \kappa$ ...
Arvid Samuelsson's user avatar
7 votes
1 answer
458 views

Reinhardt's ultimate classes

In the preface to Sets and Classes by Muller, several research programs are outlined that were in development concurrently with publication (or finished slightly beforehand) that he would have liked ...
Alec Rhea's user avatar
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2 votes
0 answers
220 views

Why may club Berkeley cardinals not be Berkeley?

"Large Cardinals beyond Choice" makes the following definitions: $\delta$ is a Berkeley cardinal if for every transitive set $M$ such that $\delta \in m$ and every $\eta \lt \delta$ there ...
Arvid Samuelsson's user avatar
4 votes
0 answers
109 views

What's the consistency strength of this strengthening of weakly superstrong cardinals?

Recall that a cardinal $\kappa$ is weakly superstrong if, for every $A \subseteq V_\kappa$, there is a cardinal $\lambda$ and a set $A^* \subseteq V_\lambda$ such that $\langle V_\kappa, \in, A \...
Arvid Samuelsson's user avatar
1 vote
0 answers
179 views

Thinning chains of elementary extensions

I'm bumping this question, since I'm still curious regarding the answer but this question seems to have gone unnoticed. Bumped again. This is a follow-up to this question, regarding a stronger variant ...
Binary198's user avatar
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1 vote
1 answer
158 views

How do chains of elementary extensions compare to shrewdness?

I thought of the following large cardinal axiom, extending the notion of $\theta$-upliftingness: Let $\eta$ be be an ordinal, and $X$ be a class of ordinals. $\kappa$ is called $\eta$-iteratively ...
Binary198's user avatar
  • 704
1 vote
1 answer
173 views

What's the consistency status/strength of this limitation principle?

$\DeclareMathOperator\iCard{iCard}$In a prior posting If we limit matters what ZFC can prove, would that be consistent? to MO, I tried to capture the informal principle of whatever ZFC proves, it is, ...
Zuhair Al-Johar's user avatar
5 votes
1 answer
180 views

Equivalences of $\mathcal{F}$-Mahloness

Taken from Math Stack Exchange. Let $\mathcal{F}$ be a set of $\mathcal{L}_\in$-formulae, $\kappa$ be a cardinal and $A \subset \textrm{Ord}$. Then, $\kappa$ is called $\mathcal{F}$-Mahlo if $A \cap \...
Binary198's user avatar
  • 704
1 vote
0 answers
96 views

Are those two theories equivalent?

Lets denote any set that is "the set of all strictly smaller subsets of it" as size- unreachable. Formally: $$\operatorname {size-unreachable}(X) \iff \\\forall Y (Y \in X \iff Y \subset X \...
Zuhair Al-Johar's user avatar
7 votes
1 answer
529 views

Shelah's "Can you take Solovay's inaccessible away?"

I was wandering if there was a book, thesis or some notes where Shelah's argument for $\mathtt{ZF}+\mathtt{DC}+$"All sets of reals are Lebesgue measurable" is equiconsistent with $\mathtt{...
Lorenzo's user avatar
  • 2,134
4 votes
1 answer
227 views

$\mathtt{PSP}$ implies the consistency of inaccessible cardinals

I'm looking for the proof that $\mathtt{PSP}$, the statement that every uncountable subset of the the Baire space $\mathbb{N}^\mathbb{N}$ contains an homeomorphic copy of the Cantor space $2^\mathbb{N}...
Lorenzo's user avatar
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5 votes
0 answers
1k views

What is the most "Icarus" Icarus set axiom?

We call a set $X ⊆ V_{λ+1}$ an Icarus set if there is an elementary embedding $j : L(X, V_{λ+1}) ≺ L(X, V_{λ+1})$ with $\mathrm{crit}(j)< λ$. But this raises the question: What is the most "...
Ember Edison's user avatar
3 votes
1 answer
285 views

Why does the second smallest worldly cardinal believe the smallest worldly cardinal is worldly?

It is known that the property of being a worldly cardinal is not absolute (a cardinal $\kappa$ is worldly iff $V_{\kappa} \vDash \textsf{ZFC}$). See here and here for more. This said, it is the case ...
aidangallagher4's user avatar

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