[FOM] intuitionism and the liar paradox

[Daniel Méhkeri]

Panu Raatikainen writes:

> Very interesting, again. But now we need to explain why it has been so  
> common among intuitionists to insist it must be decidable, and why  
> exactly they are mistaken.
>
> My reflection about decidability was just a further worry.
>
> My central worry about circularity is this: if mathematical truth is  
> explicated as provability, which means just that it would be possible  
> to prove the relevant statement, one can then ask exactly what kind of  
> notion of possibility is assumed here; and the plausible answers seem,  
> one way or other, to lean on the notion of mathematical truth. And  
> that would certainly be circular...
>
> Perhaps this worry can be circumvented - but so far, I haven't seen  
> any convincing answers...

Carl Mummert recently warned us about the adjective "constructive" 
having many meanings. I suppose this applies to a lesser extent to
"intuitionistic".

If truth is provability, and the proof relation is decidable, then
every sentence is semi-decidable. This is the so-called 
Brouwer-Kripke scheme. It states, for any sentence phi,

 (exist alpha:2^N) phi iff ( (exist n:N) alpha_n = 1 )

It is an interesting hypothesis to consider: that _in principle_, we
might have some seriously non-recursive and impredicative insight (impredicative because phi might quantify over 2^N). Still, I recall Frank Waaldijk claiming it wasn't generally accepted even by intuitionists 
in the Brouwerian tradition.

Section 2 of Charles McCarty's 1987 paper "Variations on a thesis: 
intuitionism and computability" attacks the idea that the proof relation is decidable from an "intuitionistic" perspective. 

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.ndjfl/1093637648

But his intuitionism is more like recursive constructivism, in fact
one of his points is basically that once you let go of the idea that the 
proof relation is decidable, you pretty much wind up with 
Church's Thesis. 

Regards,
Daniel Mehkeri




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