Questions tagged [large-cardinals]

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Can proper classes have different sizes?

I'm presently working in a non-ZF set theory, where there are proper classes. (Think MK or VNBG.) And I'm interested in how to think about the possibility (or impossibility) of proper classes with ...
Anonymous grad student's user avatar
4 votes
0 answers
130 views

Higher-order equivalence of ordinals

I wonder which ordinals are second-order equivalent, and similarly for other logical equivalences. Let the signature be fixed and include only <. For concreteness, let us first ask for the first ...
Alexey Slizkov's user avatar
9 votes
0 answers
207 views

Naive way to violate $\mathsf{SCH}$ at $\aleph_\omega$

I asked this question on MSE and got a partial answer. Shamefully I still haven't figured out myself a full answer, so I would like to ask it here. The usual way to get the failure of $\mathsf{SCH}$ ...
new account's user avatar
5 votes
1 answer
163 views

Can a generic ultrafilter over $\mathrm{NS}^+_{\omega_1}$ witness $\omega_1$ is Ramsey-like?

Suppose that $\kappa$ is an appropriate large cardinal (preferably a Woodin cardinal, but possibly something stronger) and let $G$ be a $\operatorname{Col}(\omega_1,<\kappa)$-generic filter over $V$...
Hanul Jeon's user avatar
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10 votes
1 answer
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1970 question of Reinhardt - how large is this ordinal?

On page 241 of William Reinhardt's paper "Ackermann's set theory equals ZF" (Annals of Math. Logic vol. 2, 1970), question 4.15 is the following: How large is the first ordinal $\gamma$ ...
C7X's user avatar
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2 votes
0 answers
122 views

The strongest reflection principle that does not violate covering lemmas

#-generated reflection, or Indiscernible-generation, is considered to be the strongest reflection principle that does not violate the covering lemma in L. [1] Is there a way to extend this success to ...
Ember Edison's user avatar
8 votes
1 answer
807 views

What is the least inaccessible cardinal for Tarski-Grothendieck set theory?

Let ordinal $\alpha$ be the least ordinal such that $V_\alpha\models$ Tarski-Grothendieck set theory. What position does $\alpha$ have in the hierarchy of inaccessible cardinals?
Frode Alfson Bjørdal's user avatar
12 votes
1 answer
443 views

Why do we need the comparison lemma?

An inner model is a standard transitive (proper class) structure which satisfies all the axioms of ZFC and contains all the ordinals. The simplest and most well-known inner model is Gödel’s $L$, which ...
Binary198's user avatar
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8 votes
2 answers
1k views

Follow up question: Shelah's "Can you take Solovay's inaccessible away?"

In this answer to the question " Shelah's "Can you take Solovay's inaccessible away?" " the following is stated: Assume that $\aleph_1$ is not inaccessible in $L$, hence a ...
C_M's user avatar
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Is $\kappa \rightarrow [\kappa]^2_3$ the same as $\kappa \rightarrow [\kappa]^2_2$ for inaccessible $\kappa$

The principle $\kappa \rightarrow [\kappa]^2_\alpha$ states that whenever we have a coloring $c:[\kappa]^2\rightarrow \alpha$ there is $H \subset \kappa$ of size $\kappa$ s.t. $|c"[H]^2|<\alpha$. ...
Jiachen Yuan's user avatar
3 votes
2 answers
307 views

Ultrafilter projections and critical points of factor maps

Suppose $j : V \to M$ is $\lambda$-supercompactness embedding derived from an $\kappa$-complete normal ultrafilter $U$ on $P_\kappa(\lambda)$, $\lambda$ regular. Suppose $\eta$ is an ordinal such ...
Monroe Eskew's user avatar
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7 votes
1 answer
537 views

What can be the measure of a Vitali set?

Suppose the continuum $\mathfrak{c}$ is real-valued measurable, i.e., there exists a countably additive probabilistic measure on $\mathfrak{c}$ that measures all subsets. Then by the construction on p....
new account's user avatar
8 votes
1 answer
592 views

Concept of bedrock and mantle in the multiverse view in the philosophy of mathematics

To be clear, I am not a mathematics educated student and I can not follow the details of the technicality of the forcing extension, but I feel that I have a good understanding of the big picture (of ...
Arian's user avatar
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23 votes
4 answers
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A Löwenheim–Skolem–Tarski-like property

I am interested in the following Löwenheim–Skolem–Tarski-like property. Given a cardinal $\kappa$, what (if any) is a property $\phi(x)$ such that if $\phi(\kappa)$ holds, then we can prove the ...
Nai-Chung Hou's user avatar
13 votes
1 answer
1k views

Are Berkeley cardinals easier to refute in ZFC than Reinhardt cardinals?

Kunen showed that Reinhardt cardinals are inconsistent in ZFC. But his proof is a bit technical for a non-set-theorist to follow. Berkeley cardinals are stronger than Reinhardt cardinals. You can ...
Tim Campion's user avatar
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12 votes
0 answers
343 views

Can Friedman's property fail at or above a supercompact cardinal?

If $\kappa>\omega_1$ is a regular cardinal, $FP_\kappa$ is the assertion that every stationary subset of $\kappa$ consisting of ordinals of countable cofinality has a closed subset of order type $\...
Ben Goodman's user avatar
6 votes
2 answers
251 views

How can we control the cardinality of $j(\kappa)$ for $\kappa$ an $\aleph_1$-strongly compact cardinal?

The following question was posted to MathStackExchange (original here). As there were no comments/answers on the original, I have ported it unedited. I am interested in determining the cardinality of $...
Calliope Ryan-Smith's user avatar
5 votes
1 answer
231 views

Collapsing every cardinal outside the Prikry sequence

All variants of Prikry forcing with collapses that i have been able to find preserve some points outside of the generic sequence (at least the successors). This is done for two reasons, (1) to obtain ...
Hannes Jakob's user avatar
  • 1,602
5 votes
0 answers
180 views

Is Vopěnka's principle inherited by Grothendieck topoi?

I call the Vopěnka's principle: Every subfunctor of an accessible functor is accessible but other formulations (which may lose equivalence in weak contexts?) are also interesting to me. If this is ...
Arshak Aivazian's user avatar
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0 answers
136 views

Does the consistency of a large cardinal axiom imply the $\omega$-consistency of that axiom?

Let $P$ be some large cardinal property (or indeed any first-order formula in the language of set theory, but lets focus on large cardinals for now). Does the $\omega$-consistency of $\mathsf{ZFC}+P$ ...
Calliope Ryan-Smith's user avatar
5 votes
0 answers
125 views

Is the opposite of the category of $\kappa$-Lindelöf Hausdorff spaces locally presentable?

Gelfand duality tells us that the category of compact Hausdorff spaces (with continuous maps as morphisms) is contravariantly equivalent to the category of commutative, unital $C^\ast$-algebras (with $...
Tim Campion's user avatar
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16 votes
2 answers
637 views

Operations on the set of large cardinal axioms

Here's a question from a non-set-theorist, but a sometime-user of large cardinals. The name Cantor's attic is pretty evocative for the collection of large cardinal axioms: looking through the pages ...
Tim Campion's user avatar
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8 votes
0 answers
330 views

Has there been any progress on this open problem about co-well-poweredness of accessible categories?

On the relations between accessible categories and large cardinal axioms, one big example is the following: Assume the existence of a proper class of strongly compact cardinals. Then every accessible ...
interregno's user avatar
5 votes
2 answers
513 views

Embedding large countable ordinals into the complex plane

Consider large countable ordinals (e.g. $\epsilon_0$ which is not "large", but still interesting). These are countable sets, so they inject into the complex plane ( or even the real line). ...
0x11111's user avatar
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1 vote
0 answers
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How much information do we need to guess a large cardinal?

Suppose $\kappa$ is a cardinal and we want to guess if $\kappa$ is a large cardinal, and if so what kind, by looking at the large cardinal status of a selection of cardinals below $\kappa$. The ...
Erin Carmody's user avatar
-1 votes
1 answer
314 views

Can MK+"Ord is almost-huge"+MM$^{++}$ be new standard foundations instead of ZFC?

I'll try to explain what this looks like to a non-expert in set theory. First, $MK$ is just a second-order $ZFC$, and there are moments when we would like to use second-order statements, for example, ...
PaleChaos's user avatar
7 votes
0 answers
385 views

What is the evidence for and against the HOD conjecture?

I'm aware that the HOD conjecture is implied by the Ultimate-L conjecture, but I don't know what the evidence is for the Ultimate-L conjecture. On the other hand, I'm aware the evidence against the ...
Someone211's user avatar
8 votes
2 answers
879 views

Large cardinal near inconsistencies

I am looking for examples of results about large cardinals, large cardinal axioms, or other objects of high (or seemingly high) consistency strength that are almost inconsistencies. I am looking for ...
Joseph Van Name's user avatar
4 votes
0 answers
218 views

stating large cardinal axioms in ZF

Can I ask whether there is a good reference for how to state the standard large cardinal axioms in the context of $ZF$? My concern is that it seems that the usual proof that embeddings defined from ...
Rupert's user avatar
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1 vote
0 answers
55 views

Cardinality of maximal elements in terms of set-theoretic inclusion in the space $c_0(\mathbb{N})$

Let $c_0(\mathbb{N})$ be the space of real-valued sequences converging to zero. Here, each element $\{\alpha_n\} \in c_0(\mathbb{N})$ itself is a "set", so that we can think about set-...
Isaac's user avatar
  • 2,727
3 votes
0 answers
188 views

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
472 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
0 answers
257 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
246 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
366 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
  • 2,774
2 votes
0 answers
188 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
  • 1,176
4 votes
1 answer
255 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
470 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 ...
Anindya's user avatar
  • 371
9 votes
1 answer
260 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
223 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 ...
Anindya's user avatar
  • 371
7 votes
1 answer
249 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
  • 1,602
2 votes
0 answers
85 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
  • 733
3 votes
0 answers
147 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
  • 704
8 votes
0 answers
170 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
114 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
221 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
242 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
65 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
232 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
256 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
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