FOM: CH and 2nd-order validity

[Kanovei]

> Date: Wed, 18 Oct 2000 01:21:41 +0200
> From: Robert Black <Robert.Black at nottingham.ac.uk>
> Subject: Re: FOM: CH and 2nd-order validity
 
> Personally, I think that every sentence of arithmetic has  a truth-value,
> and that CH has a truth-value too. but the problem is to persuade
> non-believers.
 
The problem is first of all to substantiate the assertion 
that begins with "I think", even for sentences of arithmetic, 
let alone CH. 
Presumably you mean MATHEMATICAL truth value, rather than 
something implied by any platonistical/philosophical/theological 
arguments. 
In other words you seem to "think" that any sentence of arithmetic 
(also CH) is either mathematically true or mathematically false. 

But this immediately runs into contradiction with the following 
observations, that hardly can be questioned: 

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.  

V.Kanovei