Considering what you wrote in your slide presentation "On the definitional character of axioms.", you might be interested in the following preprint by John L. Bell (found on his Homepage) titled "SETS AND CLASSES AS MANY". In it, he initially gives the following naive definition of set (following Cantor in his book, Contributions to the Founding of the Theory of Transfinite Numbers):
Set theory is sometimes formulated by starting with two sorts of entities called individuals and classes, and then defining a set to be a class as one, that is, a class which is at the same time an individual...If on the other hand we insist--as we shall here--that classes are to be taken in the sense of multitudes, pluralities, of classes as many, then no class can be an individual and so, in particular, the concept of set will need to be redefined.
What does "class which is at the same time an individual" mean? Well, following Cantor, since a set ("aggregate") is
...any collection into a whole $M$ of definite and separate objects m of our intuition and our thought. These objects are called the "elements" of $M$...In signs we express this thus: $M$={m}.
it seems reasonable to infer that a set is then a class which is a object, i.e. that which can be an element of another set (or even, possibly, of itself). This, however, produces an interesting variant of the Russell paradox, defined entirely in terms of class as object:
Is {x| x$\notin$x} both a class and object, that is, can {x| x$\notin$x} be an element of another set (including itself)? If {x| $\notin$x} is both a class and an object, a contradiction follows because then {x| x$\notin$x} can be an element of itself. On the other hand, since {x| x$\notin$x} is, in some sense a 'name' (label) of its elements, this 'name' can be an 'element' of some other collection, it must be deemed an 'object' and therefore can again be deemed an element of itself, and the contradiction again follows (perhaps the paradox ensues through a confusion between 'label' and that which is labelled, but then, can a class that is not itself an 'object' be labelled?).
Bell solves this problem in the following manner (though he does not explicitly mention the above version of the paradox):
Now while we shall require a set to be a class of some kind, construing the class concept as "class as many" entails that sets can no longer literally be taken as individuals. So instead we shall take sets to be classes that are are represented, or labelled, by individuals in an appropriate way. For simplicity we shall suppose that labels are attached, not just to sets, but to all classes: thus each class $X$ will be assigned an individual $\lambda$$X$ called its label. Now in view of Cantor's theorem that the number of classes of individuals exceeds the number of individuals, it is not possible for different classes always to be assigned distinct labels [consider now the argument put forth in Stanford Encyclopedia of Philosophy's (SEP's) entry "Frege's Theorem and Foundations for Arithmetic" (Edward Zalta's entry) that Russell's paradox is engendered because Second-order Logic+ Basic Law V requires the impossible situation in which the domain of concepts (labels) has to be strictly larger than the domain of extensions (classes) while at the same time the domain of extensions has to be as large as the domain of concepts--my comment]. This being the case, we single out a subdomain $S$ of the domain of classes on which the labelling map $\lambda$ is one-to-one. The classes falling under $S$ will be identified as sets; and an individual which is the label of a set will be called an identifier.
Bell now defines the dual notion of colabelling:
For reasons of symmetry, it will be convenient (although not strictly necessary) to assume that, in addition to the operation of labelling each class by an individual, there is a reverse process--colabelling--which assigns a class [note here that classes then, of necessity, must exist as extensions--my comment] to each individual. Thus we shall suppose that to each individual x there corresponds a unique class x* called its colabel. Again, because of Cantor's theorem, not every class can be the colabel of an individual (although every individual can be the label of a class). However, it seems natural enough to stipulate that each set be the colabel of some individual, and indeed that this individual may be taken to be the label of the set in question. Thus we shall require that $X$=$\lambda$$($$X$$)^{*}$ for every set $X$. In that event, for any identifier x in the above sense, we shall have x = $\lambda$(x*); that is, the colabel of an identifier is the set of which it is the label, or the set labelled by the identifier. Another way of putting this is to say that the restriction of the colabelling map to identifiers acts as an inverse to the restriction of he labelling map to sets.
After dealing with singletons and the empty set in the following fashion
Singletons and the empty class--"multitudes" with just one, or no members respectively--are here regarded, like the "numbers" $\mathbf 1$ and $\mathbf 0$, as "ideal" entities introduced to enable the theory to be developed smoothly.
he considers the problem of adequately defining the $\in$ relation:
The membership relation $\in$ between individuals and classes is a primitive of our system. It will be taken as an objective relation in the sense suggested, for example, by the assertion that Lazare Carnot was a member of the Committee of Public Safety, or Polaris is a member of the constellation Ursa Minor. The fact that $\in$ is not iterable--there are no "$\in$-chains"--means that it can have very few intrinsic properties. This is to be contrasted with the relation $\epsilon$ of "membership" between individuals, defined by x$\epsilon$y $\leftrightarrow$ x $\in$ y*: x is a member of the class labelled by y. This relation links the entities of the same sort and is, accordingly, iterable. It should be noted, however, that the presence of the colabelling map * in the definition of $\epsilon$ gives the latter a purely formal, arbitrary character
Next, Bell uses the $\epsilon$-relation to present the notion of nonwellfounded set. However,
In the usual set theories it is difficult to grasp the nature of a set which is, for example, identical with its own singleton since a set cannot be "formed" by assembling individuals. In the present scheme, on the other hand, the assertion $\alpha$={$\alpha$}--which is, as remarked above, not well-formed--is replaced by the assertion $\forall$$x$($x$$\epsilon$$\alpha$ $\leftrightarrow$ $x$=$\alpha$), that is, $\alpha^*$={$\alpha$}, which asserts that {$\alpha$} is identical, not with $\alpha$ itself, but rather with its colabel. Similarly, the self-membership assertion $\alpha$$\in$$\alpha$ is transformed into the statement $\alpha$ $\epsilon$ $\alpha$, that is, $\alpha$$\in$$\alpha^*$, which asserts that $\alpha$ belongs, not to itself, but merely to its colabel. And an assertion of cyclic membership $\alpha$$\in$$\mathop b$$\in$$\alpha$ is transformed into the assertion $\alpha$ $\epsilon$ $\mathop b$ $\in$ $\alpha$, or $\alpha$$\in$$\mathop b^*$& $\mathop b$$\in$$\alpha^*$, that is, "$\alpha$ (respectively $\mathop b$) is a member of the colabel of $\mathop b$(respectively $\alpha$)." These rephrasings appear much more natural in that they only impute the possession of curious properties to the colabelling map, rather than to the objective membership relation $\in$ itself.
Considering the "naturalness" of the rephrasings, and the version of the Russell paradox stated above, one might reasonably infer that the confusion between labels and colabels is at the heart of the derivation of Russell's paradox, and by making the distinction between the two, Bell has found the 'best' way of ridding systems of set theory from it.
I say this because of what Prof. Bell says in the concluding paragraph of the "Introduction" to his paper:
We shall also see that, in addition to nonwellfounded set theories, a number of other theories familiar from the literature can be provided with natural formulations within the system to be presented here [the theory $\mathbf M$ of multitudes or classes as many--my comment from the begining sentence of Bell's Section 1]. These include second-order arithmetic, the set theories of Zermelo-Frankel, Morse-Kelly, and Ackermann, as well as a system in which Frege's construction of the natural numbers can be carried out. Each of these theories can therefore be seen as the result of imposing a particular condition on a common apparatus of labelling classes by individuals [these conditions therefore define what sets 'exist' within each individual theory, a result similar to yours --my comment].
