Proposed new syntax

[Steve D'Aprano]

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.