> 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.
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 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.
> 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?
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 It's worth noting that it's a book from the late 80's however, so it's a bit outdated in some parts (he didn't know that the universe's expansion was accelerating in particular, which renders his discussion of the curvature of the universe a bit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).
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.