> 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
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
You're correct, but I think the spirit of what GP wrote is still true.
You can't capture an infinite structure fully by "finitary" methods, either you use FOL and then you run into L-S, or you use higher-order logic (for which L-S doesn't apply) but then you don't have a complete proof system anymore.
To tie it all together, L-S and incompleteness are about different flavours of "not being able to capture something". L-S is about models of different cardinalities. These models do still all satisfy exactly the same sentences. Incompleteness is about different models actually satisfying different sentences.
I'm curious about this, another comment in the thread expressed the same opinion about math pages on wiki, while I've always heard the opposite opinion among mathematicians. Could you say a bit more on what makes pretty much all of the math pages on wiki very poorly written?
They're generally written in a way where it's not helpful to teach yourself about the topic, but is helpful to refresh yourself if you've previously learned about it elsewhere.
The author is using the usual definition where vertices are an arbitrary (finite) set and edges are pairs of vertices. From the book:
Definition 2.1.1. A simple graph is a pair (V, E), where V is a finite set, and
where E is a subset of P_2(V).
Sure in some examples V happens to be a set of natural numbers and then edges must be pairs if natural numbers, but there's nothing weird with that as any set would do. (Incidentally in model theory countably infinite graphs are often assumed to have as underlying set the naturals because it is convenient notationwise)
Usually people don't explicitly post links to libgen in their public website, but the vast majority of mathematicians makes ample use of libgen and sci-hub without thinking twice
Thanks for your comment! Compiler development does sound like a very interesting area, do you have a textbook or online lectures to recommend to get started?
Unfortunately, no. The textbooks that I have looked into have not been convincing, since they tend to be overly focused on the parsing and syntax aspects, which is IMHO not where the really interesting stuff happens. (It might help if you've never had any CS exposure at all, but I did get exposed to a bunch of parsing theory in my CS minor during undergrad, mostly from a complexity perspective, so...) So I ended up picking things up on-the-fly over time, occasionally going over some presentations / lecture notes or (less commonly) papers that I found references to or found via searching.
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.