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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

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

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