Matthew P Wiener <weemba@sagi.wistar.upenn.edu> wrote: > >The evolutionary model I've described is quite constructive in its >weeding out of inconsistencies. [...] > >As such, I expect the 0-1 laws to put me in case 1 above. It is not >the case that the omega consistency is arrived at by an impenetrable >fiat, as with your example (omitted) of a case 2, but that glaringly >obvious inconsistencies are eliminated, and that given time, all >inconsistencies become obvious.
Consider the following "evolution" process starting from an alpha theory A and producing omega theory C.
Take an enumeration X(1),X(2),... of all the sentences in the language of A. Take some positive integer N. Enumerate proofs in the theory A+X(N). If a contradiction appears using axioms of A, stop. If a contradiction involving X(N), or a proof of X(N) within A appear, increment N and repeat.
The "omega" theory C is then A (if A is inconsistent) or else A+X(k) for the smallest k >= N such that X(k) is independent of A. The alpha theory A can prove this, as it can carry out Goedel's argument that Con(A) implies the existence of statements independent of A. Thus:
1. C is definable in A 2. C is recursively axiomatized, and A can prove this 3. Con(A)-->Con(C), and A can prove this 4. C is produced in a "quite constructive" way by removing inconsistencies
But there is no limit on C's strength relative to A.
