[FOM] Infinity and the "Noble Lie"

[joeshipman@aol.com]

Mycielski:
        The axiom of infinity is not so distant from reality as to claim 
that it is
nonsense (Haney), and it is not so unambiguous as to claim that it is 
true
(Shipman)....we cannot talk honestly about the truth (in the usual
sense of the word true) of any statement unless this statement refers 
in a
clear way to a real object or process (real as opposed to imaginary). 
And of
course the axiom of infinity does not refer to any such thing.

I reply:

My question is NOT about whether we can say the axiom of infinity is 
"true".

My question is about whether we can say an ARITHMETICAL statement like 
the Paris -Harrington theorem, which DOES "refer in a clear way to a 
real object or process", is "true" if we are unable to prove it from 
axioms in which we have the same degree of epistemological confidence.

A theorem cannot be MORE certain than the axioms it is derived from.  
Therefore, if you won't call the set-theoretical axiom of Infinity 
"true", you had better explain whether you are willing to call the 
Paris-Harrington Theorem "true".  If you are so willing, you should be 
able to identify "true" axioms it can be proved from.

-- JS