>>Consider the system consisting of physically sufficient mathematics.
>Okay, suppose that that's PA.
I prefer to call it "PM" = "physical mathematics", and am willing to assume it's a bit stronger than PA. It's definitely well below ZF in strength.
>>This system is good enough to carry out 0-1 arguments,
>Exactly which 0-1 arguments?
The kind that Erdos pioneered in random graph theory. The first application to logic I know of is a theorem of Fagin (in JSL), showing that first order statements about finite graphs are true with probability 0 or 1 in the random finite graph.
>>so one can explicitly define and prove consistent (as one is, in >>effect, exhibiting a model) any bounded fragment of the omega point >>of TM-mind evolution.
>I don't understand what you are saying. Are you saying that a TM-mind >produced by evolution is necessarily consistent? Why should that be?
Yes. The inconsistencies, while they have no physical effect, do mess up the minds, and thus select out. Being a deep subtle inconsistency is no help, since evolution will insert bridging axioms.
>Let's try to be more concrete. Suppose that we have a creature with a >TM-mind that believes PA. This creature has two mutant children, one >of whom believes PA + the twin primes conjecture, and the other of >whom believes PA + the negation of the twin primes conjecture. It's >not clear that PA can prove the consistency of either of these >"mutant" theories. Are you saying, then, that natural selection must >surely kill off these mutants?
No! I'm saying two things here, and you're combining them into one very wrong statement. First, natural selection will kill off mutants that believe PA+TPC+~TPC. And second, that at most one of TPC and ~TPC will be in the evolutionary omega-point. -- -Matthew P Wiener (weemba@sagi.wistar.upenn.edu)
