Here I shall present my take on this subject. To do this we need to have some intuitive background about sets, for it is in the intuitive realm that primitive concepts can be negotiated and in some way validate or otherwise rejected as ad-hoc and not ture. The main idea behind sets I think is simply to extend predicates into the object world, we speak of set of books, set of cars, set of colors, etc... also there is a perception of some hierarchy like that of Stars, Galaxies, Chains of Galaxies, etc.. those are parallels of the predicate hierarchy. What set theory is trying to do is to simply lay down which predicates can have object extensions. Or alternatively to concern itself with 'safe' predicates, i.e. those predicates that when assuming they have object extension then this assumption is assumed not to be involved with a paradox. So formally speaking we add a partial one place function symbol "e" to stand for "is the extension of", "e" takes predicate symbols (represented in upper case) as arguments. What is desired is that equivalent "predicates" have "identical" object extensions, so "equivalence" between predicates is translated into "identity" between objects extending them. So for predicates P,Q we have: eP=k ^ eQ=d ->[Ax(P(x)<->Q(x)) <-> k=d] Now membership "E" is defined after Frege as: x E y <-> Exist P (P(x) ^ y=eP) The intention is for membership relation to parallel fulfillment of predicates. Of course not all predicates can have such object extensions, that's why it is left for set theory to stipulate which kind of predicates can have such extensions. It is generally thought that a kind of *circularity* is incriminated in having absence of object extensions for some predicates. In other words a property D if assumed to have an object extension then we'll have D(eD) or we'd be having a set k that has eD as a subset of TC(k) "the transitive closure of k, i.e. the set of all elements of k and elements of those and elements of those etc..) and such that k itself fulfilling D,i.e. we have D(k), such a kind of circularity is thought to be at the root of the problem, so in order to avoid such circularity an axiom system is imposed carefully so that it shuns that circularity. An example of such system is the following one with the axioms of: [1] Non empty sets having the same members are identical. [2] For every set x there is a set of all elements of x satisfying P. [3] For every set x there is a set of all non empty sets of x [4] There is an infinite set of all empty objects. Now in this kind of set theory we have a base tier that is a set of all empty objects and this tier is infinite, the next tier is the set of all non empty sets of empty objects, the next tier is the set of all non empty sets of non empty sets of empty objects, etc... The structure of sets in this theory forbids any circularity with predicates, since a predicate that have an object extension would only hold of objects of the same tier and that would necessarily force the extension of that predicate to belong to the next tier thus the predicate cannot be fulfilled by its extension nor by any set that would have that extension as a subset of its transitive closure since it would belong to a higher tier, so this axiomatization shuns circularity altogether for that piece of predication. The above theory is equiconsistent with Simple Type Set Theory with Urelements + Infinity, i.e; TSTU+Infinity. In a similar manner the ordinary set theory of Z and ZF shuns circularity by working on well founded sets. On the other hand theories like NF and its fragments work directly on formulas expressing the predicates by shunning circularity of those formulas and thus extensions of predicates described by those formulae are regarded as safe from being involved with paradoxes, so here there is a kind of non circular comprehension, while with the ordinary set theory of Z and ZF it is thought that the structure of the sets itself prevents circular predication over them. As regards fitting set theory into a mereological base which is a desired thing really, I suggest the following line of thought. We work within General Extensional Atomic Mereology 'GEAM' Add the two place function symbol that takes objects as its first argument and predicates as its second argument the object is the fulfillment instance of P by x. We stipulate that all such instances are Mereological ATOMEs. So we can have collections of them easily by the Unrestricted Composition Principle of GEAM. We add one place predicate symbols "N" and "P" to signify "is negative" and "is positive" respectively. We define the predicate "is Ambivalent", symbolized by "Amb", as Amb(x) <-> [~P(x) ^ ~N(x)] We axiomatize that: y= -> ~[P(y) ^ N(y)] In English: every instance of fulfillment of a predicate can either be positive or negative, never both. An object extension of a predicate is defined as the Mereological collection of all fulfillment instances of that predicate over by all objects in the universe of discourse. So for example the object extension of the predicate described by the formula "~x=x" would be the collection of all fulfillment instances of that predicate from objects of the universe of discourse, Define: k=eQ <-> for all m ([m is an atom ^ m is a part of k] <-> Exist x (m=)) of course for the case where Q<->~y=y we'll have all such instances being negative in nature, thus we'd have: for all x(Q(x)<->~y=y) -> ~Exist y (y E eQ) so eQ would be the empty set. Also to be stipulated is that a fulfillment instance must be unique per its arguments, i.e. = <-> x=y ^ [Q<->D] Also to be noted that stipulating that acceptance or rejection of a predicate by an object would constitute a Mereological atom is actually justified intuitively but the problem is if an ambivalence instance is to be justified as constituting a Mereological atom, this needs to be scrutinized thoroughly. Now to further this approach either we assume that all objects extensions of predicates are Dichotomous, i.e. do not have a fulfillment instance that is ambivalent, by then we'll conclude that not all predicates have object extensions (accordingly <,> would be a partial function symbol), and then we'll be restricting ourselves to conditions of non circularity as above. The other way is to state that all predicates have object extensions but not all of those are Dichotomous! So the predicate "is an ordinal" for example would have an object extension but it won't be dichotomous, i.e.; it will have an ambivalent instance in it. This paves the way into defining a General principle of non circularity of predicates and thus isolating which ones have dichotomous extensions after avoiding that circularity, but still this is not that easy to get. This would be called the "principled approach" to the problem of paradoxes with set theory, while the above first two approaches would be called the "Controlled situation" approach which deals with a limited sector of predication. The controlled approach is more careful and thus much more trusted than the principled approach. It is worth noting that the above approach is not a multivalued logic approach, it is done in a limited kind of classical second order logic. And I also think it can be done in multi-sorted Classical first order logic as well, or in second order logic using Henkin semantics. One possible approach is to stipulate that if a predicate Q rejects x, i.e. we have ~Q(x), then N(), but if Q accepts x then either we have P() or Amb(), which one would be selected? would be left for the theory to decide, of course membership of sets must follow "positive" instances of fulfillment, i.e.; Frege's membership would be modified to the following: x E y <-> Exist Q (P() ^ y=eQ) We can also restrict predicates to those described by formulas that only use predicates E and = and where no predicate in them is quantified upon (superficially looking as a first order logic formula). A general condition of non circularity needs to be formally stipulated for the disciplined principled approach to be completed however it is still the case that one might limit speech to specific sector of dichotomous sets within that approach which would be a controlled situation like approach within the larger mainframe of the principled approach, which I'd think would give results similar to Z or ZF or their extensions. I don't have any detailed account on a successful principled general approach like that, but it is the general idea of this way bearing such potential that is worth noting. I do think this approach is fruitful and it would yield many possible treatments underneath set theory that might culminate in explaining axioms of set theory and also of providing for stronger axioms. For example one may concentrate his attention on HEREDITARILY DICHOTOMOUS SETs, add the following axioms: Axiom 1. [for all m (m E a -> m E b)] ^ a=eQ ^ b=eR -> [Q -> R]. Axiom 2. x is hereditarily dichotomous ^ Q(x) -> P(). (where Q only uses E and = as predicates) I think that axiom 2 can be further simplified to just requiring x to be dichotomous, i.e. we remove the hereditary requirement, so all dichotomous sets when fulfilling a predicate defined after a formula only using E and =, all would be having a positive fulfillment instance and thus they cause no ambivalence of fulfillment of those kinds of predicates. The two axioms do have some intuitive appeal, and they result in interpreting at least Zermelo, and I think ZFC as well!,over the realm of hereditarily dichotomous sets. Anyhow, the main point is that we want to have a safe realm of extending predicates into the object world, that's what set theory is all about. A formal account: The following is a formal system in which "set" and "set membership" are defined concepts, it is hoped that the set concept find some explication. It combines both Frege's extension of predicates theory and Mereology. However it incorporate some new concepts about fulfillment instance of object in predicates as well as the type of those instance whether they are negative or positive or ambivalent. Language: Second order logic Extra-logical primitives: One-place predicate symbols "+","-" denoted 'is positive' and 'is negative' respectively. Two-place relation symbols of "=", "P" denoting 'identity' and 'part-hood' Two place object-to-predicate function symbol "<,>" denoting fulfillment instance of .. in .. so y= is read as y is the fulfillment instance of Q by x, or equivalently the fulfillment instance of x in Q. Axioms: [1] Mereology: All first order axioms of General Extensional Atomic Mereology "GEAM" without bottom. [2] = <=> [x=y ^ Q<->R] [3] y= => y is an atom. [4] [not Q(x)] <=> -() Define "eQ": x=eQ <=> for all y((y is an atom ^ y P x) <=> Exist z(y=)) eQ is read as the "object extension of Q". Define "Set": Set(x) <=> Exist Q (x=eQ). Define "E": y E x <=> Exist Q (x=eQ ^ +()) Define "?": ?(x) <=> not [-(x)] ^ not [+(x)] ? is read as "is ambivalent". [5] Not [-(y) ^ +(y)] Define "Dichotomous": Dichotomous(x) <=> Set(x) ^ not Exist y (y is an atom ^ y P x ^ ?(y)). [6] [for all m E y (m E x)] ^ y=eQ ^ x=eR => [Q => R] [7] Dichotomous(x) ^ Q(x) => +() [where Q is expressed by a formula only using E or = as predicates]. / Theory definition finished. It is nice to see that this theory interpret Z and most possibly interpret Ackermann's set theory and thus ZFC. Zuhair 14/6/2015 Another more recent version of this method is present at the following site: http://zaljohar.tripod.com/extensions.txt For a more recent and complete version of this approach see the following site: http://zaljohar.tripod.com/explication.txt Zuhair 19/6/2015