On Tue, 22 Aug 2017 07:41 pm, Marko Rauhamaa wrote: > BTW, the main take of the metamathematical "fiasco" was that you can't > get rid of the meta-level. There's no consistent logical system that is > closed and encompasses everything including itself. You will always have > to step outside your formal system and resort to hand-waving in a > natural language.
That's not quite correct. Gödel's Incompleteness theorems only apply to "sufficiently powerful" systems. They don't apply to systems which are too weak. Not all such simple systems are either consistent or correct, but those that are, may be provable as such. Standard arithmetic is sufficiently powerful that the Incompleteness theorem applies, but not all such systems do. I've read a few people claim that disallowing multiplication from standard arithmetic renders it weak enough that you can prove it complete and correct, but since they give no proof or even evidence I have my doubts. Unfortunately the interwebs are full of people, even mathematicians, that have a lot of misapprehensions and misunderstandings of Gödel's Incompleteness Theorems. For example, there's a comment here: "It's easy to prove that ZFC is consistent in the right theory, e.g. ZFC+Con(ZFC)" https://philosophy.stackexchange.com/questions/28303/if-the-zfc-axioms-cannot-be-proven-consistent-how-can-we-say-for-certain-that-a apparently without the slightest awareness that this would be begging the question. If you assume that ZFC is consistent, of course you can prove that ZFC is consistent. This article has a very nice description of Gödel's theorems, the reactions of mathematicians to it ("outrage, condescension, bafflement, fascination, and complete disinterest"), a comparison of mathematical induction and ordinary induction, and why, ultimately, we shouldn't be too worried by the possibility that arithmetic is inconsistent: http://www.mathpages.com/home/kmath347/kmath347.htm -- Steve “Cheer up,” they said, “things could be worse.” So I cheered up, and sure enough, things got worse. -- https://mail.python.org/mailman/listinfo/python-list
