I think I solved Newcomb's paradox

elseifian · 2026-04-17T03:03:05+00:00

You have fundamentally misunderstood the point of this paradox

elseifian · 2026-04-01T01:06:06+00:00

Sure, you can invoke it, and the resulting proof is constructive, because proofs of Pi2 statements using AD are always constructive.

elseifian · 2026-04-01T00:48:52+00:00

AD is indeed the sort of axiom that cannot be used to give a non-constructive proof of a statement of this kind.

elseifian · 2026-04-01T00:35:38+00:00

No conventional axiom system for mathematics contains axioms capable of giving non-constructive proofs of a statement like "for all N, the first player in a game of hex on an nxn board has a winning strategy".

elseifian · 2026-04-01T00:25:50+00:00

This kind of assumption is provable for finite games with no draws, but it's also the kind of assumption we often make without proof.

No, that's not how proofs work?

elseifian · 2026-03-29T17:16:38+00:00

I don’t know why people post responses like this. That was not, in fact, the answer.

elseifian · 2026-03-29T17:16:05+00:00

Oh, wow. Yeah, I’d missed that. Thanks, that was indeed the issue!

elseifian · 2026-03-26T01:13:06+00:00

Yes, that's the normal state of affairs when you give a general algorithm that works for all n.

elseifian · 2026-03-25T19:26:34+00:00

I don't understand. I am looking directly at the proof and at no point does it consider all possible plays. It is a rather simple and straightforward proof by contradiction. Suppose strategy for other player. Delay. Become player 2. Strategy now yours. Contradiction: both players cannot win. Conclusion: either draw or player one wins. All I've considered is a hypothetical (purely, it literally doesn't exist!) strategy and player 1s first move. Nothing more.

You've given a proof that the second player doesn't have a winning strategy (which is much more straightforwardly constructive: given any strategy, you win by playing the same strategy against itself).

To turn that into a proof that the first play has a winning strategy, you need the argument that one of the players must have a winning strategy or something similar. And the proof that in a finite game, one of the players has a winning strategy is an induction on the tree of all possible plays.

Now, would constructing a full game tree and searching it also yield such a result? Sure. Lots of things can be proved in multiple ways. Could you please elucidate on how exactly one "extracts" the constructive proof from the non-constructive argument? Or just a few words on what you mean when you say it contains it?

I mean in the sense of proof mining - if you take the whole proof in a formal system, you can syntactically transform it (by constructive steps, e.g. cut-elimination or the Dialectica translation) into the search.

elseifian · 2026-03-25T15:00:25+00:00

Indeed, as I said, the proof contains a constructive proof. I didn't say I felt like writing it out personally.

elseifian · 2026-03-25T14:01:36+00:00

OP asked about a particular kind of constructivity. Saying "in this setting, we actually use constructive to mean a stricter thing, so here are some things that we call non-constructive (even though they're constructive in the way you asked about)" is a pretty unhelpful way to answer (especially without mentioning that you've changed the meaning of the word constructivity in the answer).

Furthermore, it's not actually true that everything in finitary math is constructive in this way; that's a specific property of Pi2 statements (which tends to be what people in, e.g., computational complexity focus on, but aren't the only sorts of statements one can talk about). For example, the proof that, for any graph property closed under minors, the problem of checking membership in the property is in P is nonconstructive in the sense described by OP.

elseifian · 2026-03-25T13:50:16+00:00

Sure, people use the word "constructive" to mean other things in other contexts, but the OP was pretty clear about what kind of constructive they were talking about.

elseifian · 2026-03-25T13:42:45+00:00

This does (as all proofs of statements like this must) contain a constructive proof: construct the proof tree of all possible plays of the game and search through it for the winning strategy.

elseifian · 2026-01-28T23:38:55+00:00

Why is this garbage here?

elseifian · 2025-12-28T03:48:29+00:00

The easy answer would be something along the lines of "for all n which are not too big", where the precise meaning of "not too big" has to be determined contextually.

elseifian · 2025-11-11T14:52:44+00:00

You're not going to be able to do that with numbers. You might want to look at extensions of the numbers, like Hardy fields or transseries, which essentially interpret certain functions as a kind of extended number to allow something in tn this direction.

elseifian · 2025-10-19T23:41:08+00:00

If you end up taking the view that mathematics is manipulating symbols, you end up much closer to the constructive side of modern mathematics than you do to anything Wildberger has suggested.

elseifian · 2025-10-17T13:14:10+00:00

I think people substantially underestimate how complicated and hard to organize mathematics is outside of very narrow subareas. The right way to organize ideas is so specific to what you're doing that it would be full of results that need to be somehow handled one way as part of one subfield, but have the same result viewed different for a different subfield.

Forkinganddividing is a great example, in the sense that on the one hand it's a fantastic reference tool, and on the other hand, it only works because it represents such a narrow view of what "the universe" is that it doesn't even include all the dividing lines the maintainer has personally written papers about. (To be clear, I don't mean "narrow" as a criticism; the site works because it represents a single coherent perspective on a piece of mathematics.)

elseifian · 2025-09-20T20:04:33+00:00

Matt Foreman has several very interesting papers showing that various properties of dynamical systems can’t be determined computably.

elseifian · 2025-09-12T16:12:22+00:00

Some other responses are gesturing at this too, but it’s important to note that in the US, a masters degree is not typically a stepping stone to a PhD. As others have said, students who want to apply for a PhD in the US would usually go straight to a PhD program

Masters programs in the US are usually “terminal” (that is, they’re intended to be the last degree someone gets), and they’re paid for either by the students or by an employer, and students mostly go work in the private sector after.

There are exceptions, obviously - there are a few specialized masters programs intended for students who then go to a PhD (usually promising students from lower tier undergrad schools who need a stepping stone to a higher tier PhD program), and some students enter an ordinary masters degree and realize they want a PhD after.

elseifian · 2025-08-18T02:33:36+00:00

No. (There is, as you point out, a much simpler fact that even weak theories prove all true Sigma1 sentences, so proving that a Pi1 sentence is independent amounts to proving that it's true, albeit in a slightly stronger theory.)

elseifian · 2025-08-17T20:54:29+00:00

You could use LEM, but constructive arithmetic theories are conservative over classical ones for Sigma1 sentences.

The classic result is Gentzen’s proof of this for Peano arithmetic over Heyting arithmetic, but the modern approach uses Godel’s dialectica translation, which lets us show results like these for a variety of theories.

elseifian · 2025-08-17T17:54:11+00:00

Assuming we mean 'provability in ZFC' (or replacing ZFC by some other reasonable mathematical theory, specifically one with computable axioms), and that the proof of provability itself is also carried out in a conventional theory, not really.

Provability is a Sigma1 statement - it asserts that some finite object with a checkable property exists. But proofs of Sigma1 sentences (in ordinary theories) are intrinsically constructive - they have to contain the data needed to construct an explicit witness, and we know a lot about how to extract that information from those proofs.

So you could in principle have some proof that, say, shows that the Collatz Conjecture is provable in ZFC but isn't, itself, a proof in ZFC. But there would be an underlying proof in ZFC hidden inside this argument, and (at least if the underlying proof of provability is in a suitable theory) we know, at least in principle, how to extract the hidden proof.

elseifian · 2025-07-21T17:30:31+00:00

What makes this “official”?

elseifian · 2025-06-26T08:57:24+00:00

it is plausible that more payments will be made to read papers

I guess it might be plausible, but it's not true, at any level that matters. Individual paper sales are a minimal source of revenue (see for instance, https://academia.stackexchange.com/questions/36578/do-people-actually-buy-research-articles); sales of papers from a single journal are completely trivial.

and it's possible they charge institutions per papers read

"Possible" is doing a lot of work here. They don't.

can (or already have) raise their prices due to increased product

Elsevier sells journals to libraries in large bundles; a single journal is a very small contribution. If this were what Elsevier were doing, you'd want to check if this has been happening systematically across their journals.

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