[FOM] CH and mathematics

[joeshipman@aol.com]

I should probably add to my remarks below that, according to my 
"necessary condition", the Continuum Hypothesis seems unlikely to be 
"definite". I am sympathetic to Woodin's view that "if CH is definite, 
then CH is false", but I haven't seen good arguments for CH's 
definiteness.

My "sufficient condition" says that pi^0 statements are definite, and 
also that any provable statement is definite. Pi^1 statements have an 
asymmetry, in that false ones are refutable and hence definite. So, to 
find a number-theoretic candidate for indefiniteness, we should look at 
Pi^2.  The twin primes conjecture and "P does not equal NP" are much 
more generally believed to be true than false. Can anyone suggest a 
natural example of a pi_2 statement of number theory for which there is 
no strong general opinion about its truth or falsity?

-- JS


*********************
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.
***


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