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*To*: fom@math.psu.edu*Subject*: FOM: RVM vs V=L*From*: JoeShipman@aol.com*Date*: Mon, 28 Jun 1999 01:11:55 EDT*CC*: friedman@math.ohio-state.edu, shipman@savera.com*Sender*: owner-fom@math.psu.edu

FOMers may find the following e-mail exchange I had with Harvey yesterday of interest: ********************************************** Shipman: I am looking at ways to augment ZFC that have strong consequences in ordinary mathematics. There are a number of well-known independent statements (GCH, Suslin Problem, Whitehead Problem, etc) that are settled by V=L; but V=L doesn't have anything new to tell us about absolute statements. I prefer to assume that c has a real-valued measure (or, slightly stronger, that c is a real-valued measurable cardinal). This has lots of consequences in both analysis and arithmetic (including all your independent finite statements and my strong Fubini theorems). My question is: are any reasonable statements of ordinary mathematics or descriptive set theory known to be independent of "c is a real-valued measurable cardinal"? The best I can think of is Projective Determinacy which is equiconsistent with cardinals well above measurables; but is there any reasonable statement about analytic sets or below which is known not to be settled by "c is RVM"? Friedman: I don't know of any reasonable statements about analytic sets or below that does not follow already from the existence of sharps, which is of course not really mathematical, but is a weak consequence of RVM. The exceptions are some statements I formulated about functions on and sets of rationals on the FOM a while ago. They go up to very very large cardinals. Any two analytic but not Borel sets of reals are Borel isomorphic is equivalent to sharps. You could use that as an axiom?! Shipman: That's a very nice statement -- but I'm not ONLY interested in statements about analytic sets or below, and RVM (which can be given some intuitive motivation) is much stronger (in particular settling CH and much more). In this sense ZFC+RVM is much "more complete" than ZFC+V=L. All it takes to intuitively motivate ZFC+RVM is to give up the geometric intuition of the uniformity and flatness of space, which modern physics makes dubious anyway. (The only problem with RVM is that the measure can't be translation-invariant.) Friedman: I have discussed in talks at Princeton Phil. Dept such merits of RVM, and the fact that set theorists do not discuss it very much in terms of being an axiom. Obviously, they don't accept it since if they did, they wouldn't regard the continuum hypothesis as an open question. Yet I don't hear much about why it is not a good axiom. If I recall, it contradicts Martin's Axiom, however, and so the set theorists may be bothered with that. I think that RVM actually is a good example of where set theoretic realists and Platonists are somewhat challenged. If they can't get a philosophical hold on the mere *status* of such a thing as RVM as an axiom - well, maybe their subject doesn't make as much well defined sense as they represent. Shipman: > I have discussed in talks at Princeton Phil. Dept such merits of RVM, and > the fact that set theorists do not discuss it very much in terms of being > an axiom. Obviously, they don't accept it since if they did, they wouldn't > regard the continuum hypothesis as an open question. Yet I don't hear much > about why it is not a good axiom. If I recall, it contradicts Martin's Axiom, > however, Of course it does -- MA implies that all sets of reals with cardinality < c have Lebesgue measure 0 so you can have a subset of the plane that's measure 0 on vertical lines and co-measure 0 on horizontal lines, while RVM allows you to prove my strong Fubini theorems. > and so the set theorists may be bothered with that. Why? Do they think MA is more plausible or do they find MA easier to work with? >I think that RVM >actually is a good example of where set theoretic realists and Platonists are >somewhat challenged. If they can't get a philosophical hold on the mere >*status* of such a thing as RVM as an axiom - well, maybe their subject >doesn't make as much well defined sense as they represent. I don't understand your point. It's not set theoretic realism and Platonism that get into trouble, the problem is simply that most of the set theorists out there right now don't like RVM for (in my opinion) bad reasons (it's easier to prove things with MA, or RVM is unfashionable because it goes all the way back to Ulam, or it can't be proved consistent [this didn't used to be a bad reason, but by now they all believe measurables are consistent, and it's wimpy to only consider new axioms like MA that have no additional consistency strength]). "The current crop of set theorists, who are realists and Platonists, neglect RVM" doesn't imply the subject isn't well-defined. A *good* reason to reject RVM would be an intuitive argument that it is false, but that the real-valued measure couldn't be translation-invariant is not such an argument. It's almost a case of sour grapes -- since our beautiful well-behaved Lebesgue measure can't be extended nicely, it can't be extended at all. The set theorists may have an intuition that sets smaller than c in cardinality ought to have Lebesgue measure 0, which contradicts RVM, but I dealt with this in my thesis when I said "It is interesting to contrast Theorem 2, which implies that if Lebesgue measure can be extended to {\it all} sets of reals then a strong Fubini theorem holds, with Sierpinski's example which shows that if Lebesgue measure is already defined for all {\it small} sets of reals then the strong Fubini theorem {\it fails}. In other words, if Lebesgue measure is already defined on too many sets, then it cannot be extended to all sets. This is not as counterintuitive as it might appear. Vitali's famous construction of a nonmeasurable set used the translation-invariance of Lebesgue measure; any measure on {\it all} sets of reals will not be invariant, so having too many sets Lebesgue measurable might well preclude a noninvariant extension." ************************************************************* By the way, the FOM posts Harvey refers to are his "positive postings" 17 and 23 from 4/26/98 and 11/6/98. The "Strong Fubini theorems" I refer to say that iterated integrals of nonnegative functions are equal whenever they exist (the classical Fubini theorem says that for measurable functions on product spaces, iterated integrals exist and are equal no matter what order the integrations are done in; my strong versions apply to arbitrary functions and I showed them to be consistent and independent and to follow from RVM in my thesis, "Cardinal Conditions for Strong Fubini Theorems", Transactions of the AMS October 1990). -- Joe Shipman

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