[01J]A distinction is made between an informal set theory and a formal set theory. 1
Informal set theory exploits all notions previously listed, but does not investigate the fundamentals, that is, the axiomatization. For this approach we recommend the text [ 11 ] ; or [ 24 ] for a brief discussion.
The most widely used formal set theory is the Zermelo–Fraenkel axiomatic, that we will shortly recall in next Section. See Chap. 6 in [ 13 ] (for a brief introduction [ 26 ] can also be fine).
In Zermelo—Fraenkel’s axiomatic set theory, all variables represent sets, so variables do not have a meaning of truth or falsehood. For this reason, in the definitions [00G] and [00Q] of well-formed formula changes the concept of ”atom”. A An atom is now a formula of the form \(a∈ b\) that has truth/falsehood value.