32 votes

Reflection principle vs universes

I'm going to go out on a limb and suggest that the book HTT never uses anything stronger than replacement for $\Sigma_{15}$-formulas of set theory. (Here $15$ is a randomly chosen large number, and ...
Jacob Lurie's user avatar
  • 17.6k
28 votes

Does Con(ZF + Reinhardt) really imply Con(ZFC + I0)?

The answer to your question is (almost) yes (almost is because of the addition of DC to the statement). Recently Gabriel Goldberg has proved ''Con(NBG+DC+Reinhardt)$ \implies$ Con(ZFC+I0)''. ...
Mohammad Golshani's user avatar
27 votes
Accepted

Reflection principle vs universes

Reflecting on Gabe's comment on my original answer, I now think what I wrote is misleading because it conflates two separate (but related) assertions: The existence of strongly inaccessible cardinals ...
Jacob Lurie's user avatar
  • 17.6k
25 votes

On independence and large cardinal strength of physical statements

Sorry, not an answer, but too long for a comment. I am the author of the "Pitowsky's Kolmogorovian models and Super-Determinism" paper. I would reject the claim that this paper is a "philosophical ...
Jakob's user avatar
  • 662
24 votes

Silver's approach to the inconsistency of $\mathrm{ZFC}$

There is rumor that Silver's efforts started as an attempt to come up with a flawless argument showing the main "theorem" in Jensen's "A modest remark." That main "theorem" says that in ZF, there is ...
Ralf Schindler's user avatar
24 votes
Accepted

Large categories vs. $\mathrm{U}$-categories: why is the loss of category-theoretic information inessential?

Let me try to answer as a set theorist, rather than as a category theorist, since I think that your question concerns at bottom a matter often considered in set theory. Namely, the essence of your ...
Joel David Hamkins's user avatar
24 votes

Is the theory Flow actually consistent?

I am one of the authors of this preprint. Concerning the Grothendieck Universe, my first answer is this. Some terms of Flow are called ZF-sets. Through the use of our restriction axiom we are able to ...
Adonai Sant'Anna's user avatar
23 votes

Does ZF+AD settle the original Suslin hypothesis?

No. Because of silly reasons. Recall that the powers of $\sf AD$ are quite limited to the world below $\Theta$. In particular, the proof that $\sf AD$ does not imply countable choice goes through ...
Asaf Karagila's user avatar
  • 38.2k
23 votes
Accepted

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

Work of Usuba combined with work of Woodin shows that if there is a Reinhardt cardinal $\kappa$ that is a limit of Lowenheim-Skolem cardinals, then there is a forcing extension in which $\kappa$ ...
Gabe Goldberg's user avatar
22 votes

On independence and large cardinal strength of physical statements

In a recent result with Shay Moran, Pavel Hrubes, Amir Shpilka and Amir Yehudayoff, we show that the answer to a basic question in statistical machine learning is determined by the value of the ...
Shai Ben-David's user avatar
22 votes

Completeness number of ultrafilters

Any countably incomplete ultrafilter has completeness number $\aleph_0$, and if $\kappa$ is measurable then any $\kappa$-complete non-principal ultrafilter on $\kappa$ has completeness number $\kappa$....
Andreas Blass's user avatar
21 votes

Reflection principle vs universes

I'd like to mention something that I think hasn't been pointed out yet. The original question began with In set-theoretic language, one fixes some strongly inaccessible cardinal $\kappa$... This ...
Mike Shulman's user avatar
  • 65.1k
21 votes
Accepted

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

Yes, it is easier to refute Berkeleys than Reinhardts. There is a very simple refutation of Berkeleys in ZFC that is due to Woodin. It is part of the motivation for his contention that Berkeley ...
Gabe Goldberg's user avatar
19 votes
Accepted

Category theory from MK class theory perspective?

Morse-Kelley set theory doesn't seem adequate for all the things one would like to do in category theory. It provides a nice treatment of proper classes, so it can deal with large categories like the ...
Andreas Blass's user avatar
19 votes

Necessary use of large cardinals in mathematics

The dual of an abelian group $A$ is defined to be the group $\text{Hom}(A,\mathbb Z)$ of homomorphisms to the infinite cyclic group. As usual with such dualities, there's a canonical homomorphism from ...
18 votes

Anti-large cardinal principles

Foreman's maximality principle (the statement: any non-trivial forcing either adds a new real or collapses some cardinals) implies there are no inaccessible cardinal. Also the consistency of Magidor'...
Mohammad Golshani's user avatar
18 votes

A “paradox” about the inner model problem

Inner model theorists use the word "canonical" to explain the problem in intuitive terms, it is indeed a vague problem, though it is as precise as anything in the region of superstrong ...
Grigor's user avatar
  • 904
18 votes

A Löwenheim–Skolem–Tarski-like property

Here's a counterexample for $\kappa=\aleph_1$: let $B$ be the structure with underlying set $\mathbb{N}\sqcup\mathcal{P}(\mathbb{N})$, equipped with the usual ordering on $\mathbb{N}$ as well as the $\...
Noah Schweber's user avatar
17 votes

Category theory from MK class theory perspective?

In terms of consistency strength, Kelley-Morse set theory does not really count as an "absurdly strong" set theory, and set theorists routinely consider far stronger theories. The consistency ...
Joel David Hamkins's user avatar
17 votes
Accepted

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

Answering this question would either require refuting choiceless large cardinals or getting close to refuting Woodin's HOD Conjecture. First, if choiceless cardinals are consistent, one cannot rule ...
Gabe Goldberg's user avatar
17 votes
Accepted

A Löwenheim–Skolem–Tarski-like property

Let me improve somewhat on Farmer's lower bound. Theorem. If there is a cardinal $\kappa$ with the stated reflection property, then there are many measurable cardinals, measurable cardinals of very ...
Joel David Hamkins's user avatar
17 votes

A Löwenheim–Skolem–Tarski-like property

Here is an upper bound: Suppose $\kappa$ is $2$-fold supercompact. Then the property holds at $\kappa$. (Recall that $2$-fold supcompactness means that for each ordinal $\lambda$, there is $j:V\to M$ ...
Farmer S's user avatar
  • 8,752
16 votes

Authorship of Grothendieck universes

This is a side matter to the main question here, but I wanted to add a bit more on the history of the universe concept, since this seems to be less widely known than it deserves. Namely, universes ...
Joel David Hamkins's user avatar
16 votes
Accepted

What are examples of non-equivalent virtualizations of a large cardinal?

An important feature which separates the notion of virtual large cardinals from the related notion of generic large cardinals is that we only consider embeddings on set-sized structures. Since most ...
Victoria Gitman's user avatar
16 votes

Necessary use of large cardinals in mathematics

Recall that the character of a point in a topological space is the smallest cardinality of a local base for that point. (So, for example, "first countable" = "every point has character $\aleph_0$".) ...
16 votes

Reflection principle vs universes

OK, I spent much of today trying to figure this out by actually looking in some detail at HTT. It's been quite a ride; I have definitely changed my perspective multiple times in the process. Currently,...
Peter Scholze's user avatar
15 votes
Accepted

What is the error in this disproof of the $\Omega$-conjecture?

Woodin's theorem says that assuming the $\Omega$ Conjecture and the existence of a proper class of Woodin cardinals, the set $\mathcal V_\Omega$ of $\Pi_2$ sentences that hold in every universe of the ...
Gabe Goldberg's user avatar
15 votes
Accepted

Does the statement 'there exists a first-order theory $T$ with no saturated models' have any set theoretic strength?

Unless I'm missing something, if $\vert T\vert+\aleph_0<\kappa$ and $\kappa^+=2^\kappa$, then we can build a saturated model of $T$ of cardinality $2^\kappa$. So: If we want a countable theory ...
Noah Schweber's user avatar
15 votes

Reflection principle vs universes

If I understand correctly, you're after a statement of the form : "If something was proved in HTT using universes, it can be proved without them by restricting to some $V_\kappa$ for $\kappa$ ...
Maxime Ramzi's user avatar
  • 13.6k
15 votes

Reflection principle vs universes

Answering this question depends strongly on exactly what you want from Higher Topos Theory, because expressing high logical strength is a different goal from expressing an aptly unified logical ...
Colin McLarty's user avatar

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