[FOM] inconsistency of P
Timothy Y. Chow
tchow at alum.mit.edu
Sun Oct 2 20:42:21 EDT 2011
Harvey Friedman wrote:
>As I indicated before on the FOM, there is a proof that any given
>finite fragment of PA is consistent, using "every infinite sequence of
>rationals in [0,1] has an infinite Cauchy 1/n subsequence".
Perhaps you haven't read any of Nelson's philosophical writings. He
doesn't believe in infinity, except "potential infinity," and regards
even so-called "finitary" reasoning (let's say, PRA) as having hidden
infinitary assumptions in it. Thus it is suspect.
Here's another way to put it. You [Friedman] have suggested before that
mathematics is "essentially" Pi^0_1. For example, if someone were to
prove P != NP, then we'd immediately want a more quantitative version of
it that gives us bounds, and we'd search for a Pi^0_1 strengthening. For
Nelson, math is essentially Pi^0_0. Nelson will accept statements of the
form "T is a theorem of X" as immediately meaningful because they're
Pi^0_0. (At least, he'll accept them if the proof has feasible length; my
guess is that he'd do the old "wait 2^n when asked if 2^n exists" trick if
you asked him about large finite numbers.) But any infinitary statement T
is at best a convenient fiction for helping us find Pi^0_0 statements, or
is perhaps a shorthand for "T is provable in X."
It's not clear to me whether there is anything that Nelson would accept
as settling the consistency of first-order Peano arithmetic in the
affirmative. So when he says that "the consistency of P remains an open
problem" I think he just means that nobody has yet found an explicit proof
of a contradiction in P.
More information about the FOM