Is This Equation a Valid Definition for Set A?

[transphenomen]

Or is it too circular?

$$A = {x | x \in A}$$

 

[micromass]

I take it that you mean $$A=\{x~\vert~x\in A\}$$. This is not a good definition of a set, since it does not determine A. The problem is that every set will be a possible A. I.e. every set A willl satisfy

$$A=\{x~\vert~x\in A\}$$

Thus you have not determined A uniquely, this means that this is not a good definition of a set.

 

[xxxx0xxxx]

A={x|x is in A} iff {[for all u(u is in A iff u is in A) and A is set]or[There is no set B such that for all u(u is in B iff u is in B) and A=the empty set]}

Since the right side of the Iff is true by virtue of the tautology, x is in A iff x is in A, A={x|x is in A} is a valid but "uninteresting" definition, i.e. to define an interesting A, we must define A elsewhere with a more "interesting" axiom of existence.

 

[Hurkyl]

To phrase it in different terms, claiming your equation as an implicit definition of A doesn't work, because the equation has more than one solution for A.