[FOM] CH and mathematics
Roger Bishop Jones
rbj at rbjones.com
Thu Jan 24 03:35:01 EST 2008
On Wednesday 23 January 2008 13:16, joeshipman at aol.com wrote:
> 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.
> ***
This seems to me too strong.
I think the concept "definite" is properly applicable to arithmetic truth
as a whole (every sentence of first order arithmetic has a definite truth
value), but I doubt that all arithmetic satisfies this condition.
I would not myself tie this concept (logically) with questions about
disagreement or consensus among mathematicians. I believe that CH can be
made definite, relative to appropriate choices of semantics for first
order set theory (e.g. "full power sets"), but I doubt that mathematicians
will ever unanimously agree on which descriptions of the semantics are
sufficiently definite.
Of course "definite" is not so definite a word that we can expect everyone
to agree about its meaning!
Roger Jones
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