FOM: Do We Need New Axioms? Upcoming Panel Discussion

[JoeShipman@aol.com]

Thanks to Harvey for an excellent exposition of several views on the issue of 
whether we need new axioms.  I hope that Maddy, Steel, and Feferman will 
correct any omissions, misrepresentations, or misplaced emphasis that 
Friedman may have inadvertently made in describing their views.

The interesting thing to me is that the different reasons that Steel, Maddy, 
and Friedman have for thinking we need new axioms lead to different and even 
incompatible new axioms.  Steel would seem to approve of axioms like 
Projective Determinacy or large cardinals in their full ontological 
generality; Friedman is interested more in arithmetical axioms like 
1-Con(Mahlos) and finite (or at least discrete) combinatorial statements, 
which in some sense capture the concrete essence of large cardinals; Maddy 
seems to emphasize axioms like V=L which may have no additional consistency 
strength but settle CH as well as questions about projective sets and the 
like. 

In my opinion, axioms like V=L which say nothing about what's happening down 
in the integers (and also have nothing to say about statements that are not 
equivalent to arithmetical sentences but are still "absolute"), while of 
intellectual interest, have very little relevance to either mathematics as it 
is practiced by most mathematicians, or to mathematics as it is used by 
scientists.  

I would not draw Feferman's conclusion that CH, because it is professionally 
uninteresting, is (or might as well be) meaningless, preferring to follow 
Quine (and Putnam and Maddy) in saying that we should take our best 
scientific theories seriously, which means (in the case of Quantum Mechanics 
at least) adding enough to our ontology that CH *is* meaningful, even if it 
is not settled by our current axioms.  But I maintain that CH will only be 
settled as true or false in the context of axioms that have arithmetical or 
scientific (or both) consequences, and which can therefore be seen to be 
supported by concrete evidence in a way that statements like V=L could not be.

I would therefore like to include in the new axioms to be discussed my own 
favorite, Ulam's axiom "there exists a real-valued measurable cardinal" 
(equivalently, a countably additive extension of Lebesgue measure to ALL sets 
of reals).  Not only does this have arithmetical consequences (since by 
Solovay it is equiconsistent with a measurable cardinal), it also settles CH 
(negatively, RVM implies there is a weak inaccessible less than or equal to 
c), and many questions about determinacy etc.  

Furthermore, it has been shown to be relevant to interesting questions in 
analysis which touch on theoretical physics (see my thesis "Cardinal 
Conditions for Strong Fubini Theorems", Transactions of the AMS 10/1990, in 
which I show RVM implies that nonnegative functions for which iterated 
integrals exist cannot have them be unequal, ruling out a whole class of 
"hidden variable" quantum-mechanical theories in which measuring noncommuting 
observables corresponds to integrating in different orders).

Finally, the RVM axiom has a property considered by everybody to be desirable 
in an axiom, namely intuitive plausibility, to a greater degree than axioms 
like V=L or Projective Determinacy or the existence of n-Mahlo cardinals.  We 
know such a measure could not be translation-invariant (Vitali's example, 
choose a set of coset representatives from R/Q ), but I claim this only 
forces revision of one's intuition of the uniformity and invariance of space, 
not of one's intuition of a primordial "mass" or "volume" defined on ALL 
subsets of R^n.

-- Joe Shipman