@John: Nice answer. I looked through Blumberg's review and thoroughly enjoyed it! Another good article along the same line is Peter Koepke's "Felix Hausdorff and the Foundations of Mathematics". In it, regarding the paradoxes, Koepke quotes Hausdorff as saying (quoted from the Grundzuge), "we want to admit the naive notion of set, but observing the restrictions which cut off the way to that [the--my comment] paradox[es]." The works of Skolem, Esser, C.C. Chang, Hinnion, Brady, P.C. Gilmore, Lirbert, and others still researching Naive Set Theory come to mind....

@Francis: I definitely will. Thanks.

to exist is to be able to be defined (eg. the productive sets)). Of course those who believe in a single background universe of sets will say that the existence of larger and larger cardinals, existence to just short of inconsistency, will allow for noncomputable sets for ordinal machines. So why is there a question regarding the existence of 0-sharp, when Solovay proved its existence assuming the existence of "at least one Ramsey cardinal"?

the next question that presents itself is whether there are models of n'th-order ZFC, n>=2, that satisfy L 'is a proper subset of' HOD--if not then this is surely evidence that L=HOD). This is why I said that problem 22 of Kunen, chapter 6 is possibly a mitigating factor. On the other hand, if L 'is a proper subset of' HOD then one seems to be in the situation where if V=HOD (as I assumed in my question) then there are non-computable sets for ordinal machines which can be defined from a finite set of ordinal parameters (more on a par with ordinary recursion theory if one assumes that

@To all who commented: Thanks. I will have to recheck my copy of Kunen's book. Nevertheless, one can easily see that from the comments there are models of ZFC where L=HOD and L 'is a proper subset of' HOD (eg. L[0-sharp]). Unless one is a believer in the set theoretic multiverse, either L=HOD or L 'is a proper subset of' HOD. If L=HOD for ZFC, this puts HOD for first-order set theory (and for first-order logic) on a par with its higher-order counterparts (consider Kunen's problem 22 for chapter 6--although it speaks to L defined for n'th-order logic n>=2 rather for n'th-order ZFC

'multiverse' do they seem to choose (I am asking for the "sociological fact"....)?

@Emil: The reason that Skolem believed that ZF (ZFC) was not a "privileged logical theory" was because "Even the notions of 'finite','infinite', 'simply infinite sequence',and so forth turn out to be merely relative within axiomatic set theory [in this case, first-order ZF (ZFC)--my comment]." Do most mathematicians use ZFC as a foundational theory or merely the 'language of set theory' as a convenient (and useful) language in which to define mathematics. If in fact they use ZFC as a foundational theory, do they implicitly choose the 'set-theoretic multiverse'? Which universe in the

Also, the title of the paper I mentioned to Andres is Timothy Chow's "A Beginner's guide to forcing". I am also interested in getting the title and cite for the Hajek paper so I can look it up.

Actually, the Skolem quote should be "privileged logical theory". Sorry.

@Emil: Thanks also for your comments. Very helpful. Regarding the portion of your comment that ZFC has been successfully adopted by the mainstream mathematical community as the foundation of mathematics (is this in fact a misquote?), I refer you to Skolem's paper "Some remarks on axiomatized set theory" (van Heijenoort, pp 290-301). If Skolem's critique of Zermelo's set theory (and by extension ZFC) is correct then first-order ZFC cannot deemed a "logically privileged theory" which is necessary for ZFC to be an adequate foundation for mathematics.

Do you know of any papers on Boolean-valued models proper that attempt to do this (unless you hold that Boolean algebras with values between 0 and 1 are a type of 'fuzzy logic', in which case a paper like "The Beginner's Guide to Forcing" might be adequate)? If you do please let me know.

@ Andres: Thanks for the comment--very helpful. I took a look at the first chapter of Bell's book, hoping it would give an intuitive motivation for generalizing from the {0,1} Boolean algebra to a Boolean algebra with more than two elements but no, all it did (correctly, of course) was show that you can consistently make the generalization. I guess what I am hoping to do is to understand how to interpret the 'intermediate values' between 0 and 1 when the Boolean algebra contains elements other than 0 or 1.

I guess I don't understand why Ackermann set theory can't produce varying extensions of the set part in accordance with the multiverse view of set theory.

Is it because it uses too strong a reflection principle? Also, in your paper "The Set-Theoretic Multiverse", in Theorem 2 (Naturalist Account of Forcing", you write, "If V is a (the) universe of set theory, then there is in V a class model of the theory expressing what it means to be a forcing extension of V." Is the impossibility of using classes to produce generic extensions of the set part due to the assumption that P is a "set forcing notion" (also in Theorem 2 you state (2) ...V is a transitive proper class in the (new) universe" so should P be a 'class forcing notion'?)

I ask this because in Balcar, Pazak, and Verner's "Forcing notion and Generic Filters", they write, "Let us broaden our minds and admit that our universe is not absolute, but that there are possible extensions in which some proper semisets of our universe V become sets. A generic filter is the gate to such extensions in a similar way as (-1)^(1/2) was to the extension of R"(in fact, a few sentences earlier they also write, "One can look at a generic filter as a kind of imaginary object, in our terminology, a proper semiset."). If proper semisets do not exist in Ackermann set theory

@Professor Hamkins: Also, in Ackermann set theory, can one define the notion of semiset? I ask this because according to you, if P is a set forcing notion, any subcollection of P that is a class is already a set (a semiset is subclass of a set, a proper semiset is a subclass of a set which is not a set). So by this definition the subcollection of P which is a class is (or at least seems to be if I am understanding the notion of semiset correctly) a semiset, just not a proper semiset. In Ackermann set theory, can there exist a proper semiset?