Vaughan Pratt wrote:
Quite right. I would add to this "and satisfying the expected equations." The "nasty sets" of which Steve speaks fail to satisy such expected equations as 2^2^X ~ X. (The power set of a set is a Boolean algebra, for heaven's sake. Why on earth forget that structure prior to taking the second exponentiation? Set theorists seem to think that they can simply forget structure without paying for it, but in the real world it costs kT/2 joules per element of X to forget that structure. If set theorists aren't willing to pay real-world prices in their modeling, why should we taxpayers pay them real-world salaries? Large cardinals are a figment of their overactive imaginations, and the solution to consistency concerns is not to go there.)
I will answer you in a Popperian key: Large cardinals are falsifiable, and are not yet falsified. They may in fact be figments of our imaginations, but then why do they keep on *working*? Could be just a coincidence -- but so could all other observation; that way lies the nullification of science in general. It's an illusion, by the way, to think that you can be rid of concerns about consistency by dumping large cardinals, that you can thus achieve a priori justification for apodictic certainty. That doesn't exist even for the natural numbers; Ed Nelson is quite right on this point. As to the question of taxpayer funding, I will not attempt to justify it (I'm a libertarian in politics), but will merely note that many taxpayers probably feel that way about *all* pure mathematics.