From: Vaughan Pratt <pratt@cs.Stanford.EDU>
From: oliver@math.ucla.edu
i) If my model says there are inaccessibles,... my model is clearly better, because it has everything in it that yours does, and more besides; moreover, we haven't lost any nice features that your model might happen to have,
Consistency is not a nice feature? News to me. My model is more likely to be consistent than yours.
Careful with language. Models are not consistent or inconsistent; they simply exist or fail to exist. Consistency is a property of *theories*. If my model exists, then it is better than yours.
Furthermore mathematics has had no trouble imagining my model since 1939 when Goedel showed us how. It is *very* hard to imagine your model, so hard that not even the smartest people in the world have been able to do it in over half a century of trying.
I do not know what you might mean by this. I have no difficulty whatsoever in imagining inaccessible cardinals. If you mean I can't imagine everything that might happen *below* an inaccessible cardinal I plead guilty, and challenge you to imagine all ordinals below Aleph_1.
This is an argument for assuming the existence of everything whose nonexistence you can't actually *prove*.
No, not everything: For example I don't assume the existence of a proof of 0=1 from ZFC, even though I can't prove the nonexistence of such a proof. In my first article I tried to give an idea of why I assume one and not the other.
But that forces you to retreat every time we disprove the existence of yet another ordinal.
-------IF YOU READ NOTHING ELSE IN THE ARTICLE, READ THIS PARAGRAPH--------- Science is a continual process of assuming things that might be proved wrong, taking the chance that you may later be forced to retreat. Read "Conjectures and Refutations" by Karl Popper.
A much more stable approach would be to accept the easily imagined, and reject what is hard to imagine. Following that strategy, you only have to retreat in the face of 50-year or even 200-year earthquakes.
Truly, I think you are much overestimating the difference between ZFC and ZFC+inaccessibles. To prove a contradiction from ZFC+inaccessibles, or even ZFC+measurables, would be earthquake enough for me; but even a contradiction from ZFC would have little effect on most everyday mathematics as long as it didn't go through in, say, 2nd-order PA.
Anyway, what are we arguing about here? What exactly *is* the advantage of assuming inaccessibles?
Well for example, if inaccessibles exist then we *know* choice is necessary to prove the existence of a nonmeasurable set of reals. If measurables exist then every analytic set of reals (i.e. projection of a Borel set in the plane) is measurable.