Re: Turing vs math

From: Juergen Schmidhuber <juergen.domain.name.hidden>
Date: Tue, 16 Nov 1999 09:48:29 +0100

There seems to be some confusion concerning Goedel etc.

All formal proofs of number theory are computable from the axioms
describable by a few bits. So what about Goedel's theorem? We cannot
derive it from the axioms. But what does that mean? It just means we
can add it as an axiom. It just means the proof of Goedel's statement
requires more bits than those conveyed by the original axioms.

In other words, Goedel used additional information. In many of the UTM's
universes his theorem will be proven by those willing to do the same.

The UTM-based universes are the formally describable ones. If some
vague "set of all mathematical structures" is supposed to contain more
than that (e.g., non-computable real numbers as opposed to mere finite
proofs concerning their properties) then it cannot be a subset of the
UTM set. Neither will it be formally describable.

Juergen www.idsia.ch/~juergen
Received on Tue Nov 16 1999 - 00:51:53 PST