[FOM] Infinity and the "Noble Lie"
joeshipman at aol.com
Sat Jan 7 16:23:20 EST 2006
> Now consider the Paris-Harrington Theorem, which changes the
> Ramsey's theorem to require that the monochromatic subset S be
> large" (|S|>min(S)). All proofs of this theorem must assume the axiom
Are you saying this theorem implies the axiom of infinity?
Otherwise, what do you precisely mean by "must assume the axiom of
I mean something very simple. Any proof in ZFC of the Paris-Harrington
Theorem must, at some point, use the ZFC Axiom of Infinity. If you
remove that axiom from ZFC, the resulting system is not powerful enough
to prove the Paris-Harrington Theorem, though it is powerful enough to
prove Ramsey's Theorem.
You may, instead, choose to prove the Paris-Harrington Theorem in some
other formal system, but you will be unable to translate that proof
into ZFC without at some point using the Infinity Axiom.
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