FOM: determinate truth values, coherent pragmatism JoeShipman at
Tue Sep 5 00:49:02 EDT 2000

In a message dated 9/4/00 9:19:31 PM Eastern Daylight Time, 
friedman at writes:

<< 4. I also see the possibility that the general math community may be more
 comfortable with "there is an atomless probability measure on all subsets
 of the unit interval" than with large cardinal axioms, in that this
 involves objects that are far less abnormal than a large cardinal. Of
 course, this particular one and a measurable cardinal are known to be
 equivalent for normal mathematical purposes. (Actually this equvalence
 passes through some sophisticated set theory that is not "convenient" for
 mathematicians, and so there is a need to put things in a form that
 mathematicians will find easy to use). Of course, a byproduct of this
 probability measure is that CH is very badly false. But, when viewed as an
 approach to CH, this is seriously at odds with the approach currently taken
 by the set theorists. After all, if they like it, then the CH would have
 been viewed as being settled long long ago by them. They reject this
 probability measure as an axiom candidate. However, from the coherent
 pragmatism that I see in the math community, they may well accept this when
 shown how to use this in an easy and coherent and effective way for a
 sufficient body of attractive normal mathematics. This would be where the
 set theorists' approach  via some notion of truth and the mathematicians
 approach via coherent pragmatism would clash. >>

Harvey, I agree with most of your latest post, but I am not sure why you are 
making a distinction between set theorists and other mathematicians regarding 
approach to new axioms:  it seems to me that the set theorists are also using 
"coherent pragmatism" (though they may like to talk in terms of truth).  I 
have certainly not seen any satisfactory arguments from set theorists why the 
axiom of an atomless measure on the continuum is FALSE; though I have seen 
arguments that alternatives to this axiom (such as Martin's axiom) are 
USEFUL, I have not seen any to persuade me that those alternatives are "true".

Can any set theorists reading this who take a realist view and are of the 
opinion that the "atomless measure" axiom is actually false (rather than 
unprofitable to study) please explain the reasons for this opinion?

-- Joe Shipman

More information about the FOM mailing list