[FOM] Infinity and the "Noble Lie"

[joeshipman@aol.com]

Shipman:

> Now consider the Paris-Harrington Theorem, which changes the 
conclusion of
> Ramsey's theorem to require that the monochromatic subset S be 
"relatively
> large" (|S|>min(S)). All proofs of this theorem must assume the axiom 
of
> infinity.

Lagnese:

Are you saying this theorem implies the axiom of infinity?
Otherwise, what do you precisely mean by "must assume the axiom of
infinity"?

I reply:

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.

-- JS