Questions tagged [large-cardinals]

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Universe V = Ultimate L inside set theoretic multiverse

Good day to you all, I would like to ask a question about relation between Prof. H. Woodin V = Ultimate L and a concept of set theoretical multiverse as proposed by Prof. Hamkins. If V = Ultimate L ...
Pan Mrož's user avatar
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4 votes
0 answers
183 views

Omega logic, V = Ultimate L and hierarchy of laws collapse

I would like to ask a question about the omega conjecture and its relationship with the V = Ultimate L axiom. In his lecture Prof. Hugh Woodin have stated that "Assuming the omega conjecture, ...
Pan Mrož's user avatar
  • 171
2 votes
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228 views

Is determinacy of (some) very long open games consistent?

For $\varphi$ a first-order sentence in the language of set theory and $\kappa$ an ordinal, let $G_\varphi^{\kappa}$ be the game of length $\kappa$ in which players $1$ and $2$ alternately play ...
Noah Schweber's user avatar
7 votes
0 answers
231 views

Is this determinacy principle consistent?

Let $\mathsf{ODet}_{\omega_1}(L(\mathbb{R}))$ be the following principle ("determinacy for simple open length-$\omega_1$ games"): If $\kappa$ is any ordinal and $X\subseteq \kappa^{<\...
Noah Schweber's user avatar
8 votes
2 answers
308 views

The consistency of $\Sigma_1$-elementary embeddings $j\colon V_{\lambda+2}\to V_{\lambda+2}$ over $\mathsf{ZFC}$

One way to stratify the large cardinal hierarchy between I3 and I1 is by using second-order elementary embeddings. We may view $j\colon V_{\lambda+1}\to V_{\lambda+1}$ as a second-order embedding $j\...
Hanul Jeon's user avatar
  • 2,252
2 votes
0 answers
145 views

Some questions about the Hyperuniverse Program

The Hyperuniverse Program, founded by Sy D. Friedman, intends to produce new second-order axioms of set theory which appropriately formalize "the universe is maximal" in one of a few ways. A ...
C7X's user avatar
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4 votes
1 answer
232 views

Cofinal inconsistency

Apologies in advance if this question is obvious/not research level. Let $\preceq$ be the consistency strength relationship on theories. Working over $ZF$ or $ZFC$, is there some large cardinal ...
Alec Rhea's user avatar
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5 votes
1 answer
328 views

Large cardinals in ZF + DC + AD

The Axiom of Dependent Choice (DC) is often considered to be an "intuitive and non-controversial" version of choice used in the proofs of many theorems in Analysis. Similarly, the Axiom of ...
user42761's user avatar
9 votes
1 answer
221 views

$0^{\#}$ and self embeddings of $L_\gamma$

Let us assume that there is a non-trivial elementary embedding $j \colon L_\gamma \to L_\gamma$ and $\gamma \geq \omega_1^V$. Can we conclude that $0^{\#}$ exists? In general, it is known that if ...
Yair Hayut's user avatar
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3 votes
1 answer
177 views

Stationary vs measurable limits for large cardinals

This is a follow-up to a question I asked earlier over here. Why are stationary limits so ubiquitous when studying large cardinals? I have noticed that there appears to be a stronger limit notion that ...
user42761's user avatar
6 votes
1 answer
226 views

Proof (or reference) about the cc-ness of termspace forcing

Recall that for $P$ a forcing order and $\dot{Q}$ a $P$-name for a forcing order, the termspace forcing $T(P,\dot{Q})$ consists of minimal-rank names for elements of $\dot{Q}$, ordered by $\dot{q}'\...
Hannes Jakob's user avatar
2 votes
0 answers
72 views

Can one extend higher randomness theory to the entire analytical hierarchy under certain large cardinal assumptions?

In the "Recursion Theory" book by C.T Chong, Liang Yu, towards the end of the book they list a few "open" research areas connected to higher computability theory. One such ...
H.C Manu's user avatar
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0 answers
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Can Ackermann's reflection be proved in this reflective theory?

If we extend the language of set theory, by adding class symbols in upper case and reserve the lower cases for sets. Add axioms: Bi-sorted ID theory (see end of this posting) + The class axioms: Sorts:...
Zuhair Al-Johar's user avatar
3 votes
0 answers
119 views

Systems of elementary embeddings

Recently I've been thinking about elementary embeddings, partition cardinals, etc. as part of my never-ending quest for understanding of consistency strength :p I came up with this idea, called I* ...
Binary198's user avatar
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0 answers
136 views

What large cardinals are needed to imply projective sets have the perfect set property?

If there are infinitely many Woodins, then every projective set is determined, whence every projective set has the perfect set property (PSP). Since determinacy is such a stronger property than the ...
Kameryn Williams's user avatar
2 votes
0 answers
95 views

How to express Kunen's inconsistency, Reinhardt and Wholeness axioms, by single sentences?

Working in $\sf NBG, $ can we express the property of a class being set theoretically definable, by a single sentence? Like for example, the following way: $$\operatorname {std}(X) \iff \exists x_1 \...
Zuhair Al-Johar's user avatar
4 votes
1 answer
200 views

Ramsey-like property with order condition

I wonder if there are regular cardinals $\lambda$ and $\kappa$ such that $\kappa < \lambda \leq 2^\kappa$ and such that, consistently, the following holds: Let $c: [\lambda]^2 \to \kappa$ be such ...
Matteo Casarosa's user avatar
4 votes
0 answers
211 views

Can we have full choice prior to Reinhardt cardinals?

Working in $\sf ZF + Reinhardt \ cardinal$, can we have full choice over all stages $V_{\alpha < \kappa}$ where $\kappa$ is the Reinhardt cardinal, i.e., the critical point of the elementary ...
Zuhair Al-Johar's user avatar
1 vote
0 answers
53 views

Can this method let choiceless large cardinals be smaller than cardinals compatible with choice?

Recall question "Can we have this sequence where choice fails and returns?" Can that theory be extended with requiring the $\mathcal V_n$'s to fulfill a choiceless large cardinal extension ...
Zuhair Al-Johar's user avatar
4 votes
1 answer
203 views

Can we have this sequence where choice fails and returns?

Can we have a sequence of transitive sets $\langle\mathcal V_0, \mathcal V_1, \mathcal V_2,...\rangle$, all modeling $\sf ZF$, such that $\mathcal P(V_n) \subset \mathcal V_{n+1}$, and the cardinality ...
Zuhair Al-Johar's user avatar
2 votes
0 answers
207 views

Harvey Friedman: The expanding mind

In reference 1, Friedman writes: I discuss my efforts concerning 3 crucial issues in the foundations of mathematics that are deeply connected with the great work of Kurt Gödel. [...] B. Are there ...
user76284's user avatar
  • 1,671
2 votes
1 answer
288 views

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
8 votes
0 answers
240 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
236 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
942 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
242 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
106 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 ...
Timo's user avatar
  • 399
3 votes
1 answer
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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
9 votes
0 answers
243 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
173 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
348 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
  • 2,252
2 votes
0 answers
178 views

"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
195 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
  • 2,252
8 votes
1 answer
243 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
487 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
  • 1,509
5 votes
0 answers
168 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
180 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
2 votes
0 answers
129 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
6 votes
0 answers
196 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
  • 2,252
8 votes
1 answer
313 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
183 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
107 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
156 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
152 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
647 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
  • 79
11 votes
0 answers
200 views

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
152 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
229 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
2 votes
0 answers
101 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
359 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
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