In Does Mathematics Need New Axioms? (pdf), Solomon Feferman discusses the kinds of foundational axioms available so far for mathematics. (Foundational axioms are primarily taken to be the axioms of logic, Peano arithmetic, and ZF set theory, together with Choice – these are axioms mathematicians use to justify their mathematical reasoning in general, while other sets of axioms are just used to define particular structures.) The Peano axioms (including induction) are justified by immediate reflection, as are the axioms of logic. Most of the axioms of ZFC are also similar justified (such as the empty set, powerset, pairing, and union axioms), while the others are accepted because of their necessity for rigorizing various mathematical arguments that had already been accepted (in particular Cantor’s arguments about the existence of ever-increasing cardinal numbers, and Zermelo’s argument that the Axiom of Choice implies the Well-Ordering Principle). These last few axioms (primarily separation, replacement, infinity, and choice) are thus justified in a somewhat extrinsic way, by appeal to their consequences. However, all of these axioms in PA and ZFC are in the end accepted for primarily intrinsic reasons – the reason their necessity for these arguments is decisive is that these arguments had already been accepted, and thus the axioms had been implicitly as well. While Frege’s inconsistent system also had intrinsic justification, it turned out to be inconsistent, and ZFC was a weakening that seemed to have similar intrinsic justification and seemed likely to be consistent as well. The extrinsic justification was just to show that these principles had in fact already been used, and thus to point out that the intrinsic justification was stronger than in Frege’s case.
In the case of large cardinal axioms, Feferman has more specific things to say. For those smaller (weaker in consistency strength) than a measurable cardinal, the justification is primarily intrinsic, using a principle Feferman calls “Cantor’s Absolute”. Basically, this principle asserts that whatever closure properties the “cardinality” of the universe has are shared by some actual cardinal. This is analogous to some form of reflection principle, according to which, for any finite true theory T, some ordinal \alpha can be found such that V_\alpha satisfies T. This principle is plausible because of Gödel’s Completeness Theorem, as I suggested in the fourth paragraph of my earlier post, “Consistency and Platonism”. Any true theory must be consistent (on a platonist account), and thus there must be some model satisfying this theory by the completeness theorem. Thus, believing that the axioms of ZFC are true (in a literal platonist sense) gives one strong reason to believe in addition that there are inaccessible, Mahlo, hyper-Mahlo, etc. cardinals.
However, none of this reasoning gives evidence for the existence of measurable (or larger) cardinals. Measurable cardinals were postulated as early as 1930, before Gödel’s completeness or incompleteness theorems, by Stanslaw Ulam, suggesting that they have some inherent mathematical interest beyond their consistency strength. In addition, measurable cardinals (and many other stronger large cardinals) create much more set-theoretic structure that is of interest, and thus have some slight extrinsic justification.
Harvey Friedman has been trying to strengthen this justification by extending the work of Paris and Harrington in providing a statement that is seen to be true by mathematicians but is independent of the given foundational axioms. The Paris-Harrington theorem turns out to be independent of PA, but equivalent to the \omega-consistency of PA (the statement that if P(0), P(1), P(2),… are all provable in PA, then so is \forall x P(x) ), and Harvey Friedman has come up with further theorems that are equivalent to stronger subsystems of second-order arithmetic. However, the results he has come up with that require large cardinal axioms (even just the ones smaller than a measurable cardinal) have no intuitive mathematical justification other than the consistency of these large cardinal axioms, which is certainly not clear for the cardinals beyond measurables. Thus, Friedman’s work on giving better extrinsic justifications (proving theorems that seem to be intuitively true) seems only to have succeeded so far for subsystems of second-order arithmetic, not extensions of ZFC. (Friedman seems to have higher hopes for his Boolean Relation Theory described in his contribution to a panel discussion about new axioms in mathematics in 2000.)
Interestingly, Feferman describes the Quine-Putnam indespensability argument as a kind of extrinsic justification for axioms. I suppose this view makes most justification of physical theories extrinsic, as they are believed to the extent that they predict the observations, not to the extent that they seem plausible. However, I think it is useful to distinguish this sort of extrinsic justification from the kind advanced for large cardinal axioms beyond measurables, which merely postulate interesting additional structure beyond what has already been observed, expected, or intuited. The former kind seems much more justifiable to me than the latter. I suppose I’ll have to read more Maddy to see what she says on the matter. But at any rate, Quine-Putnam style arguments probably won’t support much beyond RCA_0 in second-order arithmetic, much less ZFC or anything larger. Maddy needs to use mathematics (and in particular, set theory) as the scientific practice to respect in order to get this high, where Quine and Putnam just wanted enough to explain our observable physical universe.
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