At 23:12 12/12/04 -0500, Jesse Mazer wrote:
Hal Ruhl wrote:
At 09:35 PM 12/12/2004, you wrote:
Godel's theorem would also apply to infinite axiomatic systems whose axioms are "recursively enumerable" (computable). But sure, if you allow non-computable axiomatic systems, you could have one that was both complete and consistent.
A complete axiomatized arithmetic would be I believe be inconsistent as supported by to Bruno' post.
No, I'm sure Bruno was only talking about recursively enumerable axiomatic systems. He said himself that the set of all true statements about arithmetic would be both complete and consistent, so if you allow non-computable sets of axioms you could just have every true statement about arithmetic be an axiom.
Yes indeed. Most books give different definition of "axiomatic" and "recursively enumerable", but there is
a theorem by Craig which shows that for (most) theories, the notion are equivalent. (See Boolos and Jeffrey for
a proof of Craig's theorem).
Also, consistency is a pure syntactical notion, at least for theories having a symbol for "falsity" or having a negation connective. A theory (or a theorem proving machine) is consistent iff there is no derivation in it of the "falsity" (or of a proposition and its negation). Now, for the important class of first order logical theories (like Peano Arithmetics, Zermelo Fraenkel Set theory, etc.) the completeness theorem of Godel (note: the completeness, not the incompleteness one!) gives that being consistent is equivalent with having a model.
But I do think, and perhaps that's related with Hal intuition (I'm not sure), that any theory which try to capture too big things will be inconsistent. Classical example is the naive idea of set which leads to Frege theory and this one was shown inconsistent by Russell. Church's logical theory based on his Lambda calculus was inconsistent, etc. What is a little bit amazing is Hal insistence that the ALL should be inconsistent. This is not an uninteresting idea, but it is a risky idea which is in need of handling with care (like in the paraconsistent logic perhaps?).
I agree also with Jesse that to explain something to someone else there is a need to find common grounds.