I am an amateur mathematician, and certainly not a set theorist, but there seems to me to be an easy way around the reflexive paradox: Add to set theory the primitive $A(x,y)$, which we may think of as meaning that $x$ is allowed to belong to $y$ and the axiom


$\forall x,\forall y, x\in y \rightarrow A(x,y)$


and modify the axiom schema for abstraction to read, given any wff $\phi(y)$ in which $x$ is not a free variable,


$\exists x,\forall y, y\in x \leftrightarrow A(y,x) \wedge \phi(y)$ 


Then if we try to construct the set $B$ of all sets not belonging to themselves, we get


$\forall x, x\in B \leftrightarrow A(x,B) \wedge x\notin x$


Then, instead of the reflexive paradox, we get


$B\in B \leftrightarrow A(B,B) \wedge B\notin B$


which is a consistent statement that implies both $B\notin B$ and $\neg A(B,B)$. Moreover, since $B$ is arbitrary, it follows that no set can be a member of itself.


Now, this all looks correct to me, but I can not believe that such a simple trick has been overlooked for over a century. So I have to believe that either its been done and I am simply unaware of it, or I've made a mistake that is staring me in the face and I just can't see it. Can someone set me straight on this?



Avoiding reflexive paradox in set theory
As The User says in the comments, you still have a problem, aesthetically at least -- in order to prevent the existence of "silly" models, you need some axiom asserting that $\in^\*$ isn't too big. As is, a model in which $\in^\*$ always holds between any $x$ and $y$ satisfies your axioms; this means that your separation axiom just asserts the existence of the empty set, and so any collection of sets containing the empty set can form a model of your axioms. In particular, your theory is now certainly consistent, since the structure consisting of a single object, interpreted as the emptyset, is a model.


This is the primary difficulty in creating a useful set theory -- not avoiding the paradoxes, but avoiding them in such a way that the resulting theory has some semantic power, so that models of the theory all share some intuitive properties. Also, we want the theory to be powerful, in the sense that any of its models interpret the rest of mathematics. These two demands are actually tied together, since one of the semantic properties we tend to demand of a set theory is that its models function well as universes for mathematics. In this case, avoiding paradoxes too easily is actually in some sense a *bad* thing -- having too many models can get in the way of interpretive power. For example, one theorem showing that ZFC is a powerful theory -- the Reflection Theorem, that asserts that for each finite fragment F of ZFC, ZFC proves the consistency of F -- can also be thought of as a near-inconsistency result: ZFC is "as close as possible" to inconsistency, in terms of what it says about its own finite fragments.


(This is not to argue that you should stop thinking about these things! I think coming up with alternate set theories is one of the best things a logician can do with their time; or at least that's how I justify it to my advisor! But it is a good idea to keep all of these things in one's mind. In particular, I recommend at the outset setting down a list of requirements you want your set theory to satisfy: is consistent relative to PA? interprets ZFC? is formulated in seven-valued infinitary logic?\* since this will guide your process.)




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\* Nobody said those demands had to be *reasonable*, after all!


