FOM: determinate truth values, coherent pragmatism JoeShipman at
Sun Sep 17 21:04:47 EDT 2000

In a message dated 9/6/00 10:37:02 AM Eastern Daylight Time, 
steel at writes:

<<  It would be most useful to have a broadest point of view about sets
 accepted by all. If different points of view arise, it will be of
 immediate practical importance to put them together appropriately, so that
 we can continue to use each other's work. I think set theorists are
 engaged in uncovering such a broadest point of view, and deciding the CH
 is the next fundamental problem along this (never-ending) road. (The
 existence of a real valued measure, which is a statement of 4th order
 arithmetic, is significantly further off.) 

Well, if you decide CH positively, you decide the existence of a real-valued 
measure negatively.

Harvey made the excellent point that propositions with plausible alternatives 
are in a very different class than axioms whose negation doesn't get you 
anywhere.  Thus "a measurable cardinal exists" is a very different animal 
from "a measurable cardinal is consistent" because V=L is a coherent 
alternative to the former but no plausible view of sets seems to contradict 
the latter.  It seems to me that when considering CH vs RVM, which are real 
alternatives to each other and can't both be true, it won't be enough to seek 
a "broadest point of view".

I repeat my earlier question from my post of September 1:

I have certainly not seen any satisfactory arguments from set theorists why 
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

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