@Professor Ehrlich: I have been reading your above-mentioned paper and find it very interesting. If one defines No in Ackermann set theory (or in Levy theory), will the versions of No definable in models of ZFC be subfields of No?

that is, will probably not occur

@Stephan: In No $Omega$={0,1,2,3...|}. Since $omega$ is definable in No, then it exists (at least in No). Is {0,1,2,3...|} mere notation unless in some sense backed up by an Axiom of Infinity? as regards Russell's paradox, R={x: x $not a member of$ x} just cannot have "___ is a member of R" or "___ is not a member of R" predicated of it. In any case, since No delimits "everything you can define in some natural way", a Russell-type paradox involving No (unless you have one in mind) will probably occur.

in the 'real' set theoretic universe of ZFC to the countable models of ZFC. I hope you find this comment intelligible

Since No as defined by Conway (unless you can find some contradiction in his formulation) seems perfectly consistent (just as defining N by assuming the existence of 0 and applying a successor function s(x) first to 0 and then to s(0), s(s(0)), etc to generate N) is consistent, it seems to me at least that No is, as Ehrlich has proven, that No is the absolute arithmetic continuum and therefore should contain the versions of No relativised to models of ZFC (that is, Conway's construction carried out in ZFC) as subfields. In this manner, No acts as P(N) does

@Stefan: I was referring to P(N), the power set of N which of course can be put in 1-1 correspondence with the reals (the way I worded that part of the sentence was very ambiguous--sorry). As regards my second comment, since it refers to the naturalistic account of forcing, I guess your question would translate into 'How can you access things that are outside the set theoretic universe?'. My point would be, do you have to translate all set theoretic notions into some axiomatic theory for the set theoretic notions in question to be valid?

I asked this question of Professor Hamkins on boolesrings but he does not think this is the case since different models of ZFC can have different models of No (?--I hope I have not misinterpreted his answer). However, since Professor Conway has successfully defined No without recourse to axiomatic systems of set theory No seems to exist 'outside' these axiomatic systems and the models that interpret and realize them which again suggests that No might contain all the 'generic reals' various accounts of forcing (especially the naturalistic account of forcing) needs to form M[G].

@Stefan: Thanks. Since you know a thing or two about forcing, does No contain all reals defined by forcing 'constructions' of generic extensions of some model M of, say ZFC? I ask this question because, at least as I understand it, (to use the example of forcing not-CH) when M is a countable transitive model of ZFC, one uses the sets of natural numbers not contained in M to form the generic sets of M[G] and in analogy to this, No might contain generic sets for models of ZFC in a naturalistic account of forcing

One can avoid Curry's Paradox in Naive Set Theory by letting its logic of be the relevant first-order logic LP# (see Jaykov Foukzon's paper "Relevant First-Order Logic and Curry's Paradox", arXiv: math.LO/0804.4818).

It should also be noted that the set-theoretic version of Curry's Paradox can be avoided by letting the logic of Naive Set Theory be the relevant first-order logic LP# (see Jaykov Foukzon's "Relevant First-Order Logic LP# and Curry's Paradox", arXiv:math.LO/0804.4818).

Let H={x| x is a current student at Hogwarts}. Is H a set? Perhaps this thought will lead to a heuristic for dealing with the paradoxes in Naive Set Theory--Consider what can correctly be predicated of what, 'correctly' meaning so as not to lead to contradictions or falsehoods or being able to derive all possible well-formed formulas.

Consider C={x|'x is a member of x' implies F} where F is any false sentence. if 'implies' is the material conditional and 'x is a member of x' is true then "'x is a member of x' implies F" is false. Can one say that there are x satisfying a falsehood? If yes then C is a legitimate set, although one would not want to derive anything from it. If 'implies' = '|--' (where relevancy might be considered) then " 'x is a member of x' implies F, where F is false implies not-'x is a member of x' and C={ }. Consider the set S={x| x is a colorless green idea}. Is S a set?

I am also considering asking the question you suggest either by editing my question to suit or asking it as a new question. Perhaps questions 2 and 3 would be better suited to philosophy.stack.exchange?

@quid: so you are basically saying Hausdorff considered 'Set Theory' not as a 'theory proper' but merely as a language in which mathematics can be formulated? If that is the case then it would seem that Unrestricted Comprehension could be construed not as an axiom but as a rule of formation for well-formed formulas and the 'paradoxical sets' as grammatical but meaningless sentences (formulas) like "Colorless green ideas sleep furiously." Is this essentially what you are saying?