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. Bruno