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.

 

[blue_raver22]

I would say that this equation is not a valid definition for Set A. It is too circular and does not provide any meaningful information about the set itself. A valid definition for a set should clearly and concisely describe the elements that belong to the set, rather than just stating that the elements belong to the set itself. Additionally, the use of the same variable (A) on both sides of the equation can lead to confusion and is not a standard practice in mathematics. In order for this equation to be a valid definition, it would need to be more specific and provide a clear understanding of what elements are included in Set A.