[FOM] Re: Freeman Dyson on Inexhaustibility

[Martin Davis]

Don Fallis wrote about Dyson invoking G\"odel's incompleteness theorem to 
proclaim "the inexhaustibility of physics".  It should be noted that Dyson 
uses this argument to deny the possibility of a "theory of everything", 
that is, a physical theory that would encompass all physical phenomena. But 
this is a confusion on Dyson's part. One might invoke G\"odel 
undecidability to infer that even if such a theory could be found, and if 
it were formalized in something like predicate calculus, then there would 
be statements in the language of the theory that could be neither proved 
nor disproved in the theory as formalized. But each such statement would be 
decided if the underlying logic were suitably strengthened (e.g., by 
permitting abstractions of higher type).

None of this would stop the theory from being indeed a "theory of 
everything"; it would only point to the difficulty of drawing consequences 
from the axiomatic foundation of the theory. This is not so different from 
difficulties with the three-body problem which no one claims shows that 
Newtonian physics fails to be a complete theory of gravitation. (Complete 
as a theory, but of course not empirically valid as the verification of 
general relativity has shown.)

Martin