FOM: CH and RVM
Joe Shipman
shipman at savera.com
Fri Feb 18 00:30:26 EST 2000
Insall restates the axiom RVM as a question (does there exist a
real-valued countably additive measure on [0,1]) and claims that CH is
of interest because it settles this question (negatively). It is just
as interesting to put it the other way, that RVM settles the question of
CH (very negatively -- the continuum must be at least as large as the
first weak inaccessible).
This is not quite "as large as logically possible", because the
continuum need not itself be weakly inaccessible, so there could be
fewer than c cardinals below c. But I am not sure it is a valid
application of Maddy's 'MAXIMIZE' principle to say there are a lot of
cardinals, because a lot of cardinals means a dearth of 1-1
correspondences between sets. This is why GCH implies the "maximizing"
principle AC -- in the presence of so many 1-1 maps, we can construct
choice functions.
RVM is a very attractive axiom because it conforms with primitive
intuitions about mass and volume. The discovery of the Banach-Tarski
paradox (there exists a finite partition of the unit ball in R^3 into
pieces which are pairwise congruent with the pieces of a partition of
two unit balls; in the more general version, any two bounded subsets
with nonempty interior are equivalent in the same way) showed that the
related intuition about the uniformity and invariance of space was
incompatible with this, but this intuition had already been called into
question by the discovery of non-Euclidean geometries and of general
relativity. Vitali's nonmeasurable set (one point from each coset of
[0,1]/Q) emphasized that it wasn't enough to abandon rotational
invariance, even translational invariance must go if you want to
preserve countable additivity. But I claim that this is no reason to
sacrifice the more basic intuition that some consistent way of assigning
"mass" to ALL subsets of space must exist. Admittedly, the intuition
was stronger before the discovery of atoms, because continuous matter
was plausible, but despite the quantum, physics still appears to require
continuous fields.
In addition to its consequences in analysis, RVM also has the great
merit of having strong consequences in arithmetic, being equiconsistent
with a measurable cardinal (Solovay).
-- Joe Shipman
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