At 12:51 -0400 25/09/2002, Wei Dai wrote:
>If we can take the set of all deductive consequences of some axioms and
>call it a theory, then why can't we also take the set of their semantic
>consequences and call it a theory? In what sense is the latter more
>"technical" than the former? It's true that the latter may require more
>computational resources to enumerate/decide (specificly it may require the
>ability to compute non-recursive functions), but the computability of the
>former is also theoretical, since currently we only have access to
>bounded space and time.
I would say the difference between animals and humans is that humans
make drawings on the walls ..., and generally doesn't take their body
as a limitation of their memory. It is also the difference between
finite automata, and universal computers: those ask always for more
memory; making clear, imo, the contingent and local character of their
space and time bounds.
>Some would argue that it's first-order theory that's misleading. See
>Stewart Shapiro's _Foundations without Foundationalism - A Case for
>Second-Order Logic_ for such an argument.
I have read and appreciate a lot of papers by Shapiro. He has edited
also the north-holland book "Intensionnal Mathematics" which I find
much interesting than its "case for Second-order Logic".
It is not very important because, as you can seen in Boolos 93, basically
the logic G and G* works also for the second order logic. Only the
restriction to Sigma_1 sentences should be substituted by a substitution
to PI^1_1 sentences. This can be use latter for showing the main argument
in AUDA can still work with considerable weakening of comp, but I think
this is pedagogically premature.