Re: Current Issues in the Philosophy of Practice of Mathematics & Informatics

[Martin Escardo <m.escardo@cs.bham.ac.uk>]

I am not sure why these questions are being asked in this list:

On 26/07/15 16:33, Patrik Eklund wrote:
> Why, for instance, is it so clear that G??del's Incompleteness Theorem is
> a "theorem" and not a "paradox"?

I am not sure what you mean by a paradox. But let me take this as a
possible interpretation: A paradox is a statement P such that both P
and not P are theorems (or, equivalently, such that P holds iff not P
holds).

As far as current mathematical knowledge goes, Goedel's Incompleteness
Theorem is just a theorem, with significant further work needed to
elevate it to the status of a paradox.

> After all, it is nothing but a bit more subtle version of the Liar
> paradox. I paradox means Fix it!, whereas a theorem means Don't
> touch!.

For comparison, in naive set theory, the set of all sets that don't
belong to themselves does lead to a paradox, corresponding to the Liar
Paradox: this set belongs to itself if and only it doesn't. Naive set
theory is inconsistent (and hence deserves its name).

In ZFC, however, the same argument *proves* that there is no set of
all sets, and no set of sets that don't belong to themselves.

It is important here that ZFC can actually formulate the question of
whether there is a set of all sets. And the answer is no.

Of course, in principle, the possibility is open that ZFC, too, has a
some paradox. This involves exhibiting a statement P and two proofs,
following strict rules of mathematical rigour, one of the statement P
and another of the statement not P. Nobody has so far managed to
exhibit such three things.

This is so hard that probably deserves a Fields Medal (followed by
immediate eviction from the mathematical community).

M.



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