FOM: CH and 2nd-order validity
kanovei at wmwap1.math.uni-wuppertal.de
Wed Oct 18 11:29:32 EDT 2000
> From Robert.Black at nottingham.ac.uk Wed Oct 18 09:39:09 2000
> Date: Wed, 18 Oct 2000 09:34:54 +0200
> >1) to be mathematically true means to have a mathematically
> >rigorous proof;
> >2) the latter means a proof in ZFC
> >(including "category theory" as a version of ZFC);
> >3) by Goedel, there are arithmetical sentences unsolvable in ZFC.
> Actually, I would deny both (1) and (2).
It is to your advantage to have a broad view on the subject,
but maybe you extend your arguments, in denying 1) and 2),
to a pedestrian like me (examples, say)
because apparently since Euclid there has been no ANY accepted
truth of a mathematical statement except those rigorously
derived (from rather self-evident axioms),
and even if something is usually taken without a proof
(say that a triagnle has just 3 angles)
this is only because of common understanding that
(or how) a proof can be generated,
and regarding 2) it is also an observable fact that ANY
ever known rigorous mathematical proof is a ZFC proof.
(There could be a ?-mark regarding NF proofs, but so far
there seem to be no any mathematical statement,
of "ordinary" mathematics at least,
provable in NF but not in ZFC.)
> I think it's true (though not provable in ZFC) that ZFC is consistent,
That ZFC is consistent is not a mathematical fact but
rather a motivational standpoint of a mathematician.
Starting any research one should believe that what
they are doing is consistent.
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