[FOM] CH and mathematics
joeshipman@aol.com
joeshipman at aol.com
Wed Jan 23 12:49:13 EST 2008
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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