FOM: CH and 2nd-order validity

[Kanovei]

> From Robert.Black at nottingham.ac.uk Wed Oct 18 09:39:09 2000
> Date: Wed, 18 Oct 2000 09:34:54 +0200
> 
> Kanovei:
> >
> >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. 

Vladimir Kanovei