No. All it really did with respect to sets was to confuse you further, apparently. Logic and set theory are just about as fundamental as you can get in Mathematics. The wiki article is attempting to describe well-formed formulae without being obtuse or obscure. To do that, the article allows some circularity in its presentation.
Don't read any more into it than that. If you were ever to take a real course in, say, mathematical logic, you might come to appreciate the difficulties dealing with a mathematical subject without the handy, pervasive tool, logic. It is a complicated and nuanced topic, and that is why it is advanced college material.
First order logic sufficient for the ZFC axioms can be rigorously defined without circularity. However, since you continue to be confused by whether ∃x Φ is an atomic formula or if ∃x is a formula at all, there is little point in you leaving the wading pool to try cliff diving.
Let's take for example what is written about atomic formula in wiki (
http://en.wikipedia.org/wiki/Atomic_formula):
In mathematical logic, an atomic formula (also known simply as an atom) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic. Compound formulas are formed by combining the atomic formulas using the logical connectives.
So, by using mathematical logic an atomic formula is the simplest well-formed formula of the logic, that if combined with logical connectives or quantifiers , provides compound formulas.
Here are the two expressions that define atomic formula (
http://en.wikipedia.org/wiki/Well-formed_formula#Predicate_logic ):
1. If t1 and t2 are terms then t1=t2 is an atomic formula
2. If R is an n-ary relation symbol, and t1,...,tn are terms, then R(t1,...,tn) is an atomic formula
The first expression is about identity.
The second expression is about relation over terms.
Platonic existence, as I define it, is a tautological existence that does not need any identity or relation in order to exist, but this tautological existence is the basis for things that are defined by using identity or relation, where these things do not have tautological existence.
So platonic existence is simpler than the atomic formulas.
If the domain of discourse is ZFC, then set is the platonic existence of ZFC that does not need any identity or relation in order to exist.
By following the notion of set's tautological existence, ∃
x is the expression of it, and it can be found, for example, within ZFC Axiom Of Infinity, as follows:
"There exists a set
x (or ∃
x)"
(this is the platonic existence, that does not need any identity or relation in order to exist (it is a tautological existence)) "such that"
(this is the non-platonic existence, that needs identity and/or relation in order to exist (it is not a tautological existence))"the empty set is a member of
x and, whenever a set y is a member of
x, then S
is also a member of
x."
So by this finer resolution about existence, set
x tautological existence (notated by the outer "{" and "}") is inaccessible to any form of existence that is not tautological existence (where this form of existence, that is also defined as no existence at all) is between the outer "{" and "}".
Traditional Mathematics does not deal with tautological existence, and as a result ∃
x (where in the case of ZFC,
x is the notion of set) is not a valid expression of its domain of discourse.
