I believe the answer is no, but Scott Aaronson on his blog just gave in interesting argument to the contrary. This is in connection with the now famous paper Undecidability of the Spectral Gap, and what it would mean for a Hamiltonian describing an actual physical system to be mathematically undecidable.

"This is a case, in my opinion, where you should have the courage of your intuitions! :) Imagine, for example, that God assured us that Goldbach’s Conjecture was independent of ZF. Even then, I would say: Goldbach’s conjecture is true in the real world, and what I mean by that, is that every even integer greater than 2 really is a sum of two primes. (Indeed, I would know that, since if there were a counterexample, then it wouldn’t be independent of ZF)... suppose you reject this viewpoint as “Platonism,” and you say: no, for me there’s no fact of the matter about anything until it’s proven or disproven in ZF. In that case, I reply: why should you even say there’s a fact of the matter about whether something is or isn’t provable in ZF, or about whether ZF itself is consistent? You seem caught in an infinite regress, where the only way out is to admit that, while you might or might not have any intuition about what transfinite sets are that’s conceptually prior to axioms, at any rate, you do have such an intuition about the positive integers".

I believe "infinite regress" is not the right word, but what Scott seems to mean is that proof constructions in ZF (or other formal theory) are essentially equivalent to arithmetic. Let's grant that God is not a deceiver. Scott's ZF fanatic appears to be in a predicament. She already accepted God's assurance that there is a fact of the matter concerning provability of the Goldbach’s Conjecture. Is she now compelled to accept that there is a fact of the matter concerning its truth as well? Is her only way out to doubt God?


EDIT 30 May 2018

I was reminded of this question when I was reading one of Aaronson's papers over the weekend. I think I now understand the argument Aaronson is making here.

Aaronson catches the ZFer on shaky ground when he notes:

In that case, I reply: why should you even say there’s a fact of the matter about whether something is or isn’t provable in ZF, or about whether ZF itself is consistent?

The problem is that independence (and consistency) results are results from mathematical logic about ZF rather than results in ZF. Therefore, if the ZFer is going to only accept results proven in ZF then how can she even think in meta-theoretic terms of independence and consistency.

Thus, the ZFer is forced to either accept the realist argument that the independence of GC implies its truth, or she must accept that God's divine declaration is meaningless.

(Original Answer)

I am having a hard time pinning down Aaronson's argument, both philosophically and logically.

Aaronson begins by arguing that the independence of GC from ZF would imply its real world truth. This is Platonism in the sense that it assumes that GC is either true or false (in the real world), amongst other things.

He then turns around and rebuts his antagonist's objection to his Platonism with a non-Platonist argument when he says "why should you even say there’s a fact of the matter about whether something is or isn’t provable in ZF". He is rejecting LEM as applied to ZF, saying there are statements S in ZF such that "S is ZF-provable or S is not ZF-provable" is not true.

I do not see the ZF fanatic as facing a predicament. I see Aaronson facing a predicament. How can he defend his Platonist argument with a non-Platonist argument.

I have probably misunderstood both the question and Aaronson's argument. My mathematical knowledge is not very deep. I am also not entirely comfortable with his claim that our intuition concerning integers is somehow stronger than our intuition concerning transfinite sets, but that's another question.

  • He can't defend the one with the other, but the goal is not a direct proof, it is a proof by contradiction by reducing the non-Platoninc position to an infinite regress. I also think it fails, because it acts as if there are only layers of relative non-Platonism, rather than complete alternatives of which Platonism is one. It forces some version of relative non-Platonism on the person who claims to hold ZF literally, and implies she really must be closer than she claims. But she is not necessarily holding any such position.
    – user9166
    Dec 17 '15 at 16:01
  • @jobermark Yes. I realised last night that all I have done is restated conifold's headline observation concerning Aaronson's argument. I'll try giving it some more thought later today, but I wouldn't hold my breath or expect anything too clever!
    – NWR
    Dec 17 '15 at 16:05
  • Hi, Nick. That's what I thought at first, but it is more subtle. For fixed even number it is provable or disprovable in ZF that it is a sum of two primes, just check sums of all smaller primes. So once God said that Goldbach is undecidable Scott says even ZFer must admit that it comes out provable ("true") in each case w/o LEM, or we would have a ZF refutation by counterexample.
    – Conifold
    Dec 18 '15 at 21:24
  • I also think that when ZFer says "that and only that is true which is provable in ZF" she means it broadly. ZF proof theory can be coded into ZF (even PA actually), so takes the question about provability in ZF as essentially a question within ZF itself, that is how she can accept God's assurance. But Scott then says, well if you accept that there are facts of the matter about proofs in ZF coded into PA, then you are already accepting that there are facts of the matter about integers expressed by PA. So you become a PA Platonist, and must admit that Goldbach is true by your own lights.
    – Conifold
    Dec 18 '15 at 21:35
  • @Conifold Yes, it is much more subtle than that. One problem for ZFer is that there are facts of the matter in ZF that only God could know. These are facts that require proofs (or counterexamples) that are so large that no human or conceivable, finite, real world computational device could ever obtain because there is not enough time before the "big rip"! Therefore, the idea of obtaining a proof in ZF is out of the question, so ZFer cannot say that there is no fact of the matter until she sees a ZF proof of it. It's a fascinating puzzle. If I come up with anything worth while I'll update
    – NWR
    Dec 18 '15 at 23:22

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