I'm not sure why Scholz brought semiotics into the mix if Hausdorff did not specifically consider semiotics (the theory of signs) in any of his work (published or unpublished) on set theory. What I find most interesting is Hausdorff considered the attempts of Zermelo and others to axiomatize set theory in order to rid it of the paradoxes (at least the Big Three-- Russel's, Burali-Forti's, and Cantor's) premature. If Sholz is correct then he must have had something to say about the paradoxes or at least an argument as to why one could 'mature'

Thanks for the comments. I was hoping that there might be someone on Math Overflow who has deep familiarity with Hausdorff's work who might be able to answer the questions. I looked through his book on set theory (the English translation) and found no comments on the paradoxes. Also, his early chapters on sets seem to me to refer to sets of (or at least could be construed to be sets of) urelements. As far as I know (perhaps I am not being enterprising enough) one cannot derive contradictions using Naive Set Theory when one is referring to sets of urelements.

One more question: Are you familiar with Dana Scott's paper "Axiomatizing Set Theory" (found in PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS, Vol 13, Part II, 1974)? I mention this because the system of axioms discussed therein might be useful in formulating the Naturalist Account of Forcing.

Also a question regarding Theorem 32 in "Set-Theoretic Geology": given that Theorem 32 and its proof can translate to a naturalist account of forcing, are you essentially saying that the union of all the W[G(i)] in Theorem 32 would not form a model of ZFC ( in other words, you cannot use forcing to create an analogue of the Cumulative Hierarchy which 'fattens' V)?

@Prof Hamkins: I was rereading your Set Theoretic Geology paper and had a question regarding Observation 31 (the Theorem you quote in your last comment to me) and its proof: Do both carry over to a naturalist account of forcing? If so, what would be the class- model analogue to the statement, "From this map, we may reconstruct z in U, which reveals all the ordinals of W to be countable, a contradiction if U and W have the same ordinals." Would the Countability Principle be used to implement the translation?

Prof. Hamkins, Is there then a property of Cohen reals that ensures that the forcing extensions containing them form a chain (I, of course want the Cohen reals I am speaking of to form a very large complete linear order--a 'real line').

(better, think of the set S={a| a=(n/0), where n is an integer}--is there an extension of the reals which contains S ( if I remember correctly, there is a mathematical trick where one allegedly proves '1=2' by implicitly dividing by zero)? Perhaps if there is such an extension (that the trick exists suggests that there 'is'(?)). Such an extension might be said to 'collapse' the integers....). Are the names for the generic elements (of whatever type) of M[G] in M loke this?

in the reals, say for example we express reals in the language of Dedekind cuts over the rationals, i.e. the name of a real is its Dedekind cut over the rationals. What you seem to be saying is that relative to the model M and its forcing extension M[G] that the 'name' (the Dedekind cut) of the Cohen real in M[G] is in M, but the 'point' (the referent the Dedekind cut in question refers to) is not--in M it is a 'gap' or to the inhabitants of M, a 'well-formed but meaningless concatenation of symbols', just like (-1)^(1/2) would be to the inhabitants of the world of the integers

@Erin, what I am trying to ask is simply this, given a model M and its forcing extension M[G], since the names for the generic sets (if I understand correctly) in M[G] are already in M, but the referents of of those names aren't (otherwise M=M[G]), are the names of the generic sets anything like the name (-1)^(1/2) in the language of the integers? For example, we know that (-1)^(1/2) has no referent in the integers (or even in the reals) but the name can be formed in the language of the integers (since (4)^(1/2) has two referents in the integers, namely 2 and -2). To take an example

(as per the formal definition of gap in the theory of linear orders)? Since it seems incoherent to say that reals that collapse cardinals and reals that don't collapse cardinals exist on a single 'real line', it would seem reasonable to say that each type of real in this case exists on a separate 'branch' of a tree that represents a partial order. If that is so then the set of names in the formal language of L would seem to represent the maximal power set (the set of all possible sets definable in the formal language of L). Is this what you are trying to say, or have I misunderstood?

"In every universe, there are classes of names of other universes"? Now consider the formal language for L, the constructable universe. Are you, in analogy with my aforementioned simple examples (if I am not stretching this analogy too far, of course) saying that the formal language of L contains the names of all possible Cohen reals, random reals, reals that collapse cardinals, essentially all possible forcing extensions of L? If so, assuming that the formal language of L is the exact syntactic replica of the universe L then what are these names without referents in L? Gaps

Here is my question: Consider, for example, the formal language of the integers which contains following rule for constructing 'names', (a)^(1/n) where a and n are integers and n is not equal to 0. With this rule one can come up with the following names (-1)^(1/2), (2)^(1/2). Obviously (-1)^(1/2) names nothing in the reals and (2)^(1/2) names nothing in the rationals although according to the rule (a)^(1/n) where a, n are integers and n not= 0 these are perfectly well-formed names. Is this the sense in which you are speaking about

@Erin: I have a question regarding the following: "In every universe, there are classes of names of other universes [forcing extensions?] with a variety of sizes of the continuum, for example. But these classes of names are not themselves the universes to which they point since if they were then the universe which dreams of them would not be coherent [I presume because then one would end up with an inconsistent theory]. It is only in the presence of a generic filter [a consistent set of sets?] that this dream class can be can become a real universe."

C=Aleph-(alpha+1) if alpha is cofinal with omega" (Cohen, SET THEORY AND THE CONTINUUM HYPOTHESIS. pg 134, Thm 4), then take the 'Universe of Interest' to be the union of L and the forcing extensions over all Alephs. The essential idea here is that the real line should consist of all possible (Cohen) reals (short of inconsistency). I'm sure there are problems with my formulation so I would appreciate any help removing the bugs. Thanks in advance for any help given along those lines.

My conception is (for better or worse) the following: 1. Begin by forming L 2. Use forcing to extend |P(omega)| to Aleph-2 without collapsing cardinals forming a new 'universe' V 3. Use forcing without collapsing cardinals to extend |P(omega)| to Aleph-3 and continuing using forcing in this manner to form a chain of forcing extensions of L for all cardinals using this theorem of Cohen as a rule, "In N (the forcing extension of L which add reals but does not collapse cardinals) C= Aleph-alpha if alpha is not cofinal with omega in M (L, or the universe previously constructed by forcing)