[FOM] Directions for Computability Theory Beyond the PureMathematical

[joeshipman@aol.com]

There is nothing to prevent a more general physical theory than we 
currently have entailing that certain physically measurable 
dimensionless constants (such as the fine-structure constant, or lepton 
mass ratios) are DEFINABLE but NONRECURSIVE real numbers. If sich a 
theory were correct, one form of Church's thesis would be false and we 
could, by an effective procedure, obtain a nonrecursive sequence (and 
thereby gain physical knowledge of mathematical facts which ZFC cannot 
prove, since any nonrecursive definable sequence has elements whose 
value ZFC does not settle).

The evidence for such "knowledge" would not be mathematical but 
physical, we would believe those mathematical facts for the same reaosn 
we believe physical theories to be correct. Godel envisioned this 
possibility.

-- JS