[FOM] CH and mathematics

[joeshipman@aol.com]

-----Original Message-----
From: Vladimir Sazonov <Vladimir.Sazonov at liverpool.ac.uk>
Please, will you explain, WHAT DOES THIS DEFINITENESS REALLY MEAN?


"Definiteness" is, I think, a primitive concept, undefinable in terms 
of prior concepts, which you are free to reject. I can give necessary 
conditions and sufficient conditions for it, though.

The following is sufficient for "A is definite":

***
One can effectively find a Turing machine T and prove that
1) Either T halts with output "True" or T halts with output "False"
2) If T halts with output "True", then A
3) If T halts with output "False, then not-A
***

The following is necessary for "A is definite":

***
Mathematicians cannot permanently disagree on the truth-value of A in 
the sense that some will insist "A is true" and others will insist "A 
is false" -- they may disagree on whether it HAS a truth-value, and 
they may disagree on whether a particular truth-value has been 
established, but at most one of the two truth values {True, False} has 
the possibility of becoming permanently accepted by a consensus of 
mathematicians.
***

-- JS

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