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Apr
19
comment Does “Every infinite set is splittable” imply $\mathsf{AC}$?
Can you give a reference for this result, or at least some pointers for finding out more? Without that, it’s pretty much just a rephrasing of the question.
Apr
17
comment What do globes (used to construct globular sets, $\omega$-categories, etc.) actually look like?
@Dominik: yes, exactly.
Mar
24
comment No matter how many algebraic invariants we attach to topological spaces, there will always be nonhomeomorphic spaces agreeing on all their invariants
@DanielMoskovich: I presume because the points here are still extremely imprecise — “infinitely varied”, “an enormous amount of structure”, “tend to be different” — and don’t even hint towards any precise statements. It takes a pretty generous reading to find the suggestion “cardinality counting” in this answer.
Mar
4
awarded  Nice Question
Mar
4
comment The universe of sets, existential quantification in set theory
One extra point to an excellent answer: von-Neumann–Bernays–Gödel set theory gives a setting much closer to ZFC than Morse–Kelly (precisely, NBG is conservative over ZFC) where one can talk directly about proper classes and so literally make a statement “the class of all sets is proper”.
Mar
4
asked Is Lemma D4.5.3 in the Elephant correct? (“In a topos, weakly projective implies internally projective.”)
Mar
3
comment What are the implications of the new quasi-polynomial time solution for the Graph Isomorphism problem?
@Aurel: Timothy Chow has updated his answer with a link to Babai’s talk at the IAS, which looks very informative!
Mar
2
comment Why should we believe in the axiom of regularity?
This sounds to me more like an argument for the anti-foundation axiom, which says exactly that when you make precise the “structure” that you get by iteratively unfolding a set, there is a unique set for any such structure. Foundation is equivalent to the restricted version that there is a unique set for any well-founded such structure.
Feb
25
comment Have there been any updates on Mochizuki's proposed proof of the abc conjecture?
@DavidRoberts: is this upcoming workshop publicly announced yet, and if so, can you point us to the announcement?
Feb
24
comment When does the projective model structure on functors exist?
@FernandoMuro: why not make that an answer? It seems like it is already a complete answer…
Feb
24
revised Finiteness Conjecture (New Doomsday conjecture)
added attribution of talks in answer
Feb
23
comment If F is left adjoint to G, when does FG preserve limits? When do counits interchange with limits?
For the benefit of other seekers: here are the paper’s pages at ResearchGate and at Elsevier’s ScienceDirect. At least from institutions with suitable journal subscriptions, the full text is available from those links, though the scan is a bit low-quality.
Feb
21
comment What's Reeb's take on naive integers?
From your last sentence, you seem to have two actual questions here: (1) “is it really true that there is no Nelson-style justification in Reeb?” and (2) “what ultimately is Reeb’s own argument?” Of these, (1) seems a clear and in principle answerable question, perfectly appropriate. But (2) seems open-ended to the point of unanswerability: if you have read Reeb’s papers and don’t feel they give a clear answer, then how can an MO answerer know what will satisfy you? I suggest it would be helpful to either remove or clarify (2).
Feb
21
comment When do powers and ends in functor categories act pointwise?
@AndreaDiBiagio: if I remember right, all the material needed appears in Mac Lane’s Categories for the working mathematician. On the other hand, the first place I remember starting to understand ends and powers (aka cotensors) over general monoidal categories was from Kelly’s Basic concepts of enriched category theory
Feb
21
answered When are subcategories of continuous functors reflective?
Feb
7
comment Widely accepted mathematical results that were later shown wrong?
@FernandoMuro: The issue (as extensively explored in Lakatos’s book) is that this formula was known “for all polyhedra” before a precise (by modern standards) definition of polyhedron had been established. So the “obvious counterexamples” were not seen as counterexamples, because they obviously weren’t polyhedra. However, when people did start exploring definitions for polyhedron, then (for some of those) this expected result became false.
Feb
4
comment Constructive compactness for countable models?
On the other hand, while constructive model theory casts its net much wider than classical model theory (including Kripke models and much more), it certainly includes ordinary “Tarski” models as a special case of these. So there is no problem with speaking of “compactness for (countable) (Tarski) models”. The reason Tarski models are less-studied constructively isn’t because they’re problematic, it’s just that there may not exist enough of them for completeness, so one is forced (no pun intended) to look at more general kinds of models.
Feb
4
comment Constructive compactness for countable models?
I don’t know the answer, I’m afraid; but unlike other commenters, I think this is a good and well-posed question. Formal constructive reverse mathematics has been investigated by e.g. Ishihara, Nemoto, and colleagues, who have certainly considered what intuitionistic formal systems are required for equivalences between WKL, LLPO, and related principles; I have heard several conference talks by them on such issues, though I don’t remember their results precisely. (cont’d)
Feb
3
comment Ref request: modelling regular theories as an injectivity condition
@ZhenLin: ah, thankyou! Henrik Forssell also pointed out to me that the same converse appears in the Adamek/Rosicky book, in the hint to Exercise 5.e. These would be close enough to serve as a reference if there’s nothing closer, but it would be nice to have this version itself if it’s been set down somewhere.
Feb
3
revised Ref request: modelling regular theories as an injectivity condition
added further info in question