[comp.ai.philosophy] Godel's Proof

G.Joly@cs.ucl.ac.uk (Gordon Joly) (11/27/90)

In article <4168@media-lab.MEDIA.MIT.EDU>, minsky@media-lab.media.mit.edu (Marvin Minsky) writes
< In other words, none of this makes sense to me.  I see Godel's theorem
< as asserting that consistency is incompatible with heuristics, in the
< sense that there is no way to ensure only true assertions in suitably rich
< systems.  Very sad, but has nothing to do with our rich, heuristic,
< fallible brains.

Godel only produced one major work; his brain was fallible and he
suffered from depression for the latter part of his life.

I have asked mathematicians of note who tell me that Godel's theorem
doesn't bother them. There is still have plenty of work to do. (But
not for the following reasons).

In a suitably rich *axiomatic* system, Godel's proof tells us to
expect to find statements we cannot prove in that axiomatic system.
One escape, in a system with N axioms, is to label that unprovable
statement axiom N+1.

Godel's proof goes on to say that N will tend to infinity; so we end
up asserting all that we cannot prove. But that is getting us into
talk.politics territory ;-)

Anyway, are heuristics axiomatic systems?

Gordon Joly                                       +44 71 387 7050 ext 3716
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Computer Science, University College London, Gower Street, LONDON WC1E 6BT

torkel@sics.se (Torkel Franzen) (11/27/90)

In article <1306@ucl-cs.uucp> G.Joly@cs.ucl.ac.uk (Gordon Joly) writes:

   >Godel only produced one major work; his brain was fallible and he
   >suffered from depression for the latter part of his life.

   This is a piece of misinformation. Godel produced a number of
results of very great importance for the development of logic. His
major contributions were 1) the completeness theorem for predicate
logic, 2) his two incompleteness theorems, 3) his definition of the
constructible hierarchy in set theory and the basic result L |=AC+GCH,
4) the ("Dialectica") interpretation of arithmetic by means of
higher order primitive recursive functions. (Add to this e.g. his
observations concerning intuitionistic logic and axioms of infinity,
and his philosophical writings.) An astonishing proportion of work in
logic and set theory directly or indirectly derives from Godel's work.

  Of course there is no reason to believe that this work is of interest
in AI. In particular, I have seen no invocation of Godel's theorem in
this context which might not as well be replaced by an invocation of the
fact that there exist non-recursive r.e. sets.