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>If a mathematical statement has no proof within ZFC we (usually) don’t call it a “truth”, regardless of if it is true or not. I'm not sure I understand what t

by simias 5y ago

>If a mathematical statement has no proof within ZFC we (usually) don’t call it a “truth”, regardless of if it is true or not. I'm not sure I understand what this changes to the overall situation. Remember that a formal system also can't prove its own consistency, so you can't just ignore bits that you don't like and expect that it doesn't matter, it's possible that one of these "true but unprovable" propositions would actually render your system inconsistent, in which case it ought to be addressed. I mean in the end Godel's own proof effectively relies on this: he builds a proposition using the rule of the system saying "this proposition has no proof" and at this point you either have to accept that the proposition is incorrect, but then your system is inconsistent (it can be used to derive falsehoods) or that it is correct but then the system is incomplete. So you may decide not to call it a truth but you also can't really call it a falsehood. I would also wager that mathematicians generally prefer incomplete to inconsistent systems, if which case you have to accept that these unprovable propositions are indeed truths. It's like the "does the set of all sets not containing themselves contain itself" paradox, you can't just ignore it as irrelevant because it puts into question your entire formal system.

Filligree 5y ago
> I'm not sure I understand what this changes to the overall situation. Remember that a formal system also can't prove its own consistency, so you can't just ignore bits that you don't like and expect that it doesn't matter, it's possible that one of these "true but unprovable" propositions would actually render your system inconsistent, in which case it ought to be addressed. Is that true? It seems to me that if a statement is unprovable, then even if it would in theory render your system inconsistent, there's no way that can actually happen. Since, if you found any example of a chain of logic which leads to an inconsistency, that would count as a proof of the original statement. Which is unprovable, and as such can't have proofs.
- simias 5y ago
  But what if the statement is "this statement can't be proven"? You haven't proven the statement, you've proven that it can't be proven. See https://youtu.be/HeQX2HjkcNo?t=926 https://youtu.be/HeQX2HjkcNo?t=926 for instance which is (IMO) a fairly good explanation of the proof and why you end up with something both true and not provable. Penrose's "Emperor's New Mind" also discusses this topic at length (within the context of whether human brains can be emulated). I wasn't convinced with his argument regarding AI but his overview of the problem of what's computable and what's not is rather good if a bit dense.
  - Filligree 5y ago
    > Penrose's "Emperor's New Mind" also discusses this topic at length (within the context of whether human brains can be emulated). I wasn't convinced with his argument regarding AI but his overview of the problem of what's computable and what's not is rather good if a bit dense. As far as I'm aware physics is computable, so I don't buy that one. :) At most one might argue that there are high-level optimizations to be had, which can't be used because they're uncomputable. In which case is it really an optimization?
    - simias 5y ago
      That's part of his arguments, that some corners of quantum physics that still elude us (quantum gravity and all that) may be undecidable. He then goes on to explain how he thinks these phenomena could have an influence on cognition. I don't really buy this theory, but the book is basically 80% "here's what we know" and 20% "here's what I conjecture" so it's still a worthwhile read IMO. It's worth noting that it's a book from the late 80's however, so it's a bit outdated in some parts (he didn't know that the universe's expansion was accelerating in particular, which renders his discussion of the curvature of the universe a bit obsolete).
- ProfHewitt 5y ago
  It turns out that foundational mathematical systems can prove their own consistency. See the following: https://papers.ssrn.com/abstract=3603021 https://papers.ssrn.com/abstract=3603021 However, the proof is not convincing because the proof is valid even if the system is inconsistent!

by simias 5y ago

Filligree 5y ago

> I'm not sure I understand what this changes to the overall situation. Remember that a formal system also can't prove its own consistency, so you can't just ignore bits that you don't like and expect that it doesn't matter, it's possible that one of these "true but unprovable" propositions would actually render your system inconsistent, in which case it ought to be addressed. Is that true? It seems to me that if a statement is unprovable, then even if it would in theory render your system inconsistent, there's no way that can actually happen. Since, if you found any example of a chain of logic which leads to an inconsistency, that would count as a proof of the original statement. Which is unprovable, and as such can't have proofs.

simias 5y ago
But what if the statement is "this statement can't be proven"? You haven't proven the statement, you've proven that it can't be proven. See https://youtu.be/HeQX2HjkcNo?t=926 https://youtu.be/HeQX2HjkcNo?t=926 for instance which is (IMO) a fairly good explanation of the proof and why you end up with something both true and not provable. Penrose's "Emperor's New Mind" also discusses this topic at length (within the context of whether human brains can be emulated). I wasn't convinced with his argument regarding AI but his overview of the problem of what's computable and what's not is rather good if a bit dense.
- Filligree 5y ago
  > Penrose's "Emperor's New Mind" also discusses this topic at length (within the context of whether human brains can be emulated). I wasn't convinced with his argument regarding AI but his overview of the problem of what's computable and what's not is rather good if a bit dense. As far as I'm aware physics is computable, so I don't buy that one. :) At most one might argue that there are high-level optimizations to be had, which can't be used because they're uncomputable. In which case is it really an optimization?
  - simias 5y ago
    That's part of his arguments, that some corners of quantum physics that still elude us (quantum gravity and all that) may be undecidable. He then goes on to explain how he thinks these phenomena could have an influence on cognition. I don't really buy this theory, but the book is basically 80% "here's what we know" and 20% "here's what I conjecture" so it's still a worthwhile read IMO. It's worth noting that it's a book from the late 80's however, so it's a bit outdated in some parts (he didn't know that the universe's expansion was accelerating in particular, which renders his discussion of the curvature of the universe a bit obsolete).
ProfHewitt 5y ago
It turns out that foundational mathematical systems can prove their own consistency. See the following: https://papers.ssrn.com/abstract=3603021 https://papers.ssrn.com/abstract=3603021 However, the proof is not convincing because the proof is valid even if the system is inconsistent!