[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

More information about the FOM mailing list