[FOM] Pluralistic Foundational Crisis?/set theory/completed

[joeshipman at aol.com]

Thanks to Toby Meadows for an excellent discussion. Perhaps the "one universe" view isn't as problematic as some suppose. Here is the way I think many mathematicians see it:


1) Consider "full" second order arithmetic. This is something we understand and we can define clearly which propositions are true and false. We can also define elementary submodels of this theory which we know do not contain all the "real real numbers", but complaints from mathematicians that, for any particular (and necessarily incomplete) axiomatization of this theory, there are propositions whose truth value we cannot derive, and that there are particular propositions of second order arithmetic whose truth value does not seem likely to be derivable from any axioms we might reasonably come to accept, would not be seen as indicating that there was not a genuine fact of the matter about the truth of any such proposition.




2) Now consider V_omega+omega, the natural model of Zermelo Set Theory with choice. It's not a model of ZFC, but it's completely nicely definable and the Continuum Hypothesis is either True or False in it. No matter how many levels above omega+omega you build on top of it, with or without inaccessibles, it's sort of obvious that the truth value of CH shouldn't be changing. If we find it natural to say that our inability even in principle to settle many questions of second order arithmetic doesn't mean they're pseudo-questions, I fail to see why we should feel required to call CH a pseudo-question.It's not even as if there are no plausible axioms that settle it in one direction or the other.


3) How "concrete" can we make CH, intuitively? Well, Aleph-1 is a concrete object, and so is Aleph-2. If you believe there exists a function from nonempty sets of real numbers to real numbers, with the property that f(X) is always an element of X, then there is no avoiding the meaningfulness of whether the unique well-ordering compatible with that function runs out before Aleph-2, or gets as far as Aleph-2. Therefore, it seems that someone saying CH is a pseudo-question with no truth value might as well be saying "the existence of a function with the property described above is a pseudo-question with no truth value".


4) So I have no argument with people who don't accept Choice for sets of reals saying that CH is a pseudo-question with no truth value. But I'm baffled how anyone who believes there is at least one choice function for sets of reals can claim it isn't meaningful whether or not the well-ordering given by that choice function peters out before Aleph-2 (obviously if there is any choice function for reals, then for the well-orderings associated with all the choice functions, either all fall short of Aleph-2, or none fall short of Aleph-2).


-- JS
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