Le 25-mai-06, à 09:04, Kim Jones a écrit :

> what would an "unreasonable machine" be like? You seem to be implying
> they exist, also that they can prove things about their possible
> neighborhoods and or histories. (?)

They are degrees. The worst "unreasonableness" of a (platonist or 
classical or even intuitionist) machine is when she believes some plain 
falsity (like p & ~p, or 0 = 1). The false implies all propositions, so 
that such machine believes everything, including everything about their 
maximal consistent extensions or histories (which does not exist). 
Those machines are just inconsistent.

Then you have machines which, although they are consistent, are not 
self-referentially correct. They are unsound, and does also believe 
some falsity, but here the falsity is irrefutable. Like consistent 
machine asserting they are inconsistent, or, curiously enough (it is a 
consequence of the second incompleteness theorem), consistent machine 
asserting they are consistent. Although it is true that they are 
consistent they cannot assert it without becoming either inconsistent, 
or, if they assert it in some special cautious way, they become 
different machine (and in that case they remain consistent and also get 
some new provability power).

But again here we anticipate. I hope I will make clear that with Church 
thesis the notion of computability will appear as absolute and 
universal, and then we will see that the notion of provability is 
relative and never universal, although some universal pattern can 
appear there too (like the logic G and G*, etc.).

I guess we will come back on this, but we have to do a transfinite 
ascension before :)



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