Paul Taylor wrote:
Now let L(infinity) be the union of L(n) over n:N.
If L(infinity) |- false then L(n) |- false for some n.
But L(infinity) |- ``L(n) is consistent,''
so L(infinity) proves its OWN consistency, contradicting Godel's theorem.
How do you conclude, from the fact that L(infinity) |- "L(n) is consistent", that L(n) is in fact consistent? Generally, if T1 |- "T2 is consistent", then to conclude "T2 is consistent", we use the following argument: Suppose T2 is inconsistent. Then there is some proof by which T2 |- false. Assuming T1 is strong enough to formalize the deductive system being used, then it follows that T1 |- "T2 is inconsistent". But by hypothesis, T1 |- "T2 is consistent", therefore T1 is inconsistent. But this is not a contradiction unless we were already *assuming* the consistency of T1 ! I.e. it follows from T1+Con(T1) that if T1 |- Con(T2), then Con(T2), but it does *not* in general follow from T1 alone. So the step from L(infinity) |- "L(n) is consistent" to L(n) is consistent cannot be formalized in any obvious way in L(infinity), and therefore you cannot (again in any obvious way) conclude L(infinity) |- "L(infinity) is consistent." -- Disclaimer: I could be wrong -- but I'm not. (Eagles, "Victim of Love") Finger for PGP public key, or visit http://www.math.ucla.edu/~oliver. 1500 bits, fingerprint AE AE 4F F8 EA EA A6 FB E9 36 5F 9E EA D0 F8 B9