ginnungagap's comments

I know Talagrand because some of his work comes up in topological dynamics (work around Rosenthal's l¹-dichotomy culminating in the Bourgain-Fremlin-Talagrand dicothomy for compact sets of Baire class 1 functions), but I had no idea he has such accomplishments in other fields! Impressive

Wow that was fast, how does that safety system work?

> This is simply false, as I already explained.

I'm sorry but this is just wrong. Since you seem to like Shapiro's book more than traditional set theory books let me quote from page 144 that the existence of an uncountable set is a theorem of ZFC: "Let C be the statement of Cantor's theorem. It entails that the powerset of the collection of finite ordinals is not countable. Since C is a theorem of first-order ZFC..."

Also this is not how the metatheory is understood in mathematics, not even in Shapiro's book, who dedicates two whole chapters to the metatheory

It is a theorem of ZFC that uncountable sets exist and every model of ZFC will have a set that the model believes to be uncountable. It doesn't matter than the metatheory might believe that model to be countable (why should the metatheory have the correct notion of what it means to be countable anyway?).

Of course, but what is your point?

Löwenheim-Skolem gives you a countable elementarily equivalent submodel (assuming you're working in a theory in a countable language, otherwise it gives you an elementary substructure of the same cardinality of the language at best), but plenty of interesting properties of familiar mathematical objects cannot be captured by a first-order theory and are not preserved by elementary equivalence, completeness of the reals being the standard example

Beginners do that exactly until the first time they have a render crash halfway through, then they learn about rendering each frame as an image (yes I learned this lesson the hard way)

If to understand D you need to know both B and C, each of which requires familiarity with A, the graph is not a tree

Those seem to be working! Thanks!

Italian here, I was trying to access anna's archive today (and libgen as well) without success, is there any easy workaround?

Oh I see, I was confused because I didn't mention PPE in my comment, maybe you meant to answer to the other top level comment which does use this abbreviation!

I'm not sure what's your point, that hardly looks like a math heavy degree

Weird to write an article about Liz Truss as a math snob (a description that seems rather stretched to be honest) without mentioning that her father, John Truss, is a fairly well known model theorist! (Even though they're not really very close from what I've been told by students of the latter)

The pdf includes famous altered images from the 2006 Lebanon war but not other well known staged ones from the same conflict, so maybe the author does consider staging a different category?

Enver Hoxha, the Albanian dictator from 1944 to 1985, also had the habit of having photos altered to remove former allies that fell out of favour and just generally making himself look better, but he's not included in this pdf. To be fair there's probably examples of historical altered photos from any dictator or sufficiently authoritative government after the invention of photography.

> I think this paper is refuting Conway (and others') proof of the claim that a set can be divided into 3+ parts without relying on the Axiom of Choice.

This paper is not refuting Conway's, and Conway's paper does not prove the claim that a set can be divided in 3+ parts without relying on AC.

What the Conway's paper proves is that, without assuming AC, if there is a bijection between A×n and B×n for some finite n, then there is a bijection between A and B. Axn can be equivalently written as the union of a×n, as a ranges over the elements of A, similarly B×n can be written as the union over b×n. This paper shows that if instead you take the union over a×N_a, where the sets N_a are pairwise disjoint and have n elements, and similarly instead of considering B×n you consider the union of b×N_b, where the sets N_b are pairwise disjoint and have n elements, then the existence of a bijection between those two unions is not sufficient to construct a bijection between A and B if we're not assuming AC. The main point here is that without choice we cannot order all of the N_a's and N_b's at the same time, while in Conway's paper, since N_a=N_b=n={0,1,...,n-1}, they are already uniformly ordered and no such issue arises.

Let (X,≤) be a partially ordered set. Define a category C whose objects are the elements of X, while for the morphisms there is a single arrow x→y iff x≤y. Those are called posetal categories and are often used as examples

> this idea isn't new, Leica did this in 2012

They also released the M10 monochrom in 2020 and the M11 just this year. As long as there's people willing to pay 7k for a (B&W) camera, leica is going to make them

> the system not being able to prove its own consistency doesn't mean that it being inconsistent!

Another funny thing that can happen is that a system proves its own inconsistency, despite being consistent. The short summary is to never trust a system talking about its own consistency.

> In other words, it shows that there is something fundamentally impossible in trying to capture an infinite structure (like numbers) by finite means (e.g. recursively axiomatizable).

Let me just point out for other readers that Löwenheim-Skolem applies to ANY first order theory (in a countable language, or also in an uncountable language if stated in the form that a theory with an infinite model with cardinality at least that of the language has infinite models in all cardinalities at least as big as that of the language), it doesn't care about how complex the axioms are from a computability point of view