Which are concepts undefined? which are symbols in the language of set theory?

Question

I know that in the most general terms, when we talk about a mathematical theory, we have in mind a collection of axioms and Undefined Concepts in a certain language.

Now in axiomatic set theory ;

• Which are concepts undefined?

• which are symbols in the language of set theory?

Answer 1

Zermelo-Fraenkel set theory ($\mathsf {ZF}$) :

is an axiom system formulated in first-order logic with equality and with only one binary relation symbol $\in$ for membership.

Thus, its language is that of first-order logic, with sentential connective symbols: $→,¬, \lor,\land$; individual variables: $v_1, v_2, \ldots$; the equality symbol $=$; and the quantifier symbols: $\forall, \exists$.

In addition to them, the theory has only one (binary) predicate : $\in$ to mean "membership". It is the only "undefined" concept of the theory.

With them we can formulate the axioms of $\mathsf {ZF}$ set theory.

One of the axioms is the Null Set axiom :

$∃x \ \forall y \ \lnot (y ∈ x)$.

Using this axiom and the Extensionality Axiom it is provable that the set satisfying the axiom is unique.

Thus, we may introduce the defined term "$\emptyset$" (a new symbol) to denote it.

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Note

The most common formulation of $\mathsf {ZF}$ set theory is based on FOL with equality.

Thus "$=$" is part of the background logic, that means that the usual axioms like $∀x(x=x)$ are assumed.

There are versions where the background logic does not include equality "$=$"; in that case $x=y$ may be defined as an abbreviation for :

$∀z[z∈x ↔ z∈y] ∧ ∀w[x∈w ↔ y∈w]$,

with a suitable reformulation of the Extensionalty Axiom. In that case, the usual properties of = must be proved from the above definition.