paradoxes and intuitionist logic
Tue, 17 Aug 1999 04:14:32 +0200
>>: Ken Evitt
> yes. i've heard of godel's theorem,
> but i'll admit i don't really understand it.
Godel's theorem is a deep result about the fact that Truth
cannot be interned within a consistent finitary system,
but that Provability can, as soon as you have computable functions.
Hence, you CANNOT write a sentence such as "this sentence is false",
but you CAN write a sentence such as "this sentence is unprovable".
> sure, one could extend the system by assigning arbitrary
> numbers to represent the system's theorems and proofs ("godel numbering"?)
> but this seems to be quite an extension to me, and it results in a new
> system that deserves to be incomplete.
Not quite. No need for of any system extension to write in it
a godel numbering of itself (yes, the choice of numbering is arbitrary).
However, of course, you need a meta-system in which to assert
that a chosen numbering indeed adequately models the system within itself
(however, you need that meta-system to talk about the system, anyway,
and to describe its finitary structure).
>> I definitely suggest you read _Godel, Escher, Bach_
Yeah, great book. Anything by D. Hofstadter is great.
> anyway, i'm still thinking on my original question (well, i guess it
> was sort of a question): what are some good ways of formulating the
> class of statements (propositions)?
Formulating it from where?
Good according to what criteria?
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