I am reading Naive Set Theorey by Halmos, and am struggling to understand the Axiom of specification.
I understand: Axiom of specification. To every set A and to every condition S( x) there corresponds a set B whose elements are exactly those elements x of A for which S( x) holds.
But I don’t understand what Halmos is talking about here: It follows that, whatever the set A may be, if B = {x ∈ A: x ∉′ x}, then, for all y, Can it be that B ∈ A? We proceed to prove that the answer is no. Indeed, if B ∈ A, then either B ∈ B also (unlikely, but not obviously impossible), or else B ∈′ B. If B ∈ B, then, by (), the assumption B ∈ A yields B ∈′ B–a contradiction. If B ∈′ B, then, by () again, the assumption B ∈ A yields B ∈ B–a contradiction again. This completes the proof that B ∈ A is impossible, so that we must have B ∈′ A. The most interesting part of this conclusion is that there exists something (namely B) that does not belong to A. The set A in this argument was quite arbitrary. We have proved, in other words, that nothing contains everything,
How can B ever belong to B?
Thanks in advance