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?

What do you mean by Undefined Concepts? and in the language of $\sf ZFC$ the only symbol is $\in$ — ℋolo, Dec 16 '18 at 12:39
There is no single answer to this. In any axiomatic theory, we may take concepts A and B to be "undefined" and define concept C in terms of those, or take B and C to be "undefined" and define A in terms of B and C. — user247327, Dec 16 '18 at 12:41
@drhab there are set theories with Urelements https://en.wikipedia.org/wiki/Urelement#Urelements_in_set_theory — ℋolo, Dec 16 '18 at 12:46
@Holo .for example in esuclidean geometry :undefined Concepts are point ,... — Almot1960, Dec 16 '18 at 12:49
@Almot1960 why not? https://en.wikipedia.org/wiki/Point_(geometry)#Points_in_Euclidean_geometry — ℋolo, Dec 16 '18 at 12:52
Oh, undefined as in "open to interpretation"(not logical symbol, part of the language)? If so only $\in$, all other objects are definable using $\in$, even $=$(in $\sf ZFC$) — ℋolo, Dec 16 '18 at 12:56
@Almot1960 x=y: $\forall z(z\in x\iff z\in y)\land \forall w(x\in w\iff y\in w)$ — ℋolo, Dec 16 '18 at 13:04
@Holo .let $x,y$ be a element . then $x \in y$ or $y\in x$ It is meaningless .because we say that $x$ is an element of $A$ (is a set no element) . — Almot1960, Dec 16 '18 at 13:09
@Almot1960 I am not sure what you mean. I defined x=y using only ∈ and logical symbols, if x,y∈A it does not mean x∈y or y∈x — ℋolo, Dec 16 '18 at 13:12
@Holo .let $A=\lbrace 1 ,2,3 \rbrace$ and $B =\lbrace 1 \rbrace $ now we want to say : "$1=1$" . How is that ? . $1 \in 1$ It is meaningless . — Almot1960, Dec 16 '18 at 13:23
@Almot1960 in $\sf ZF$ all objects are sets, 1 is usually defined as ${\emptyset}$, that is why I said $\sf ZFC$ at the end there — ℋolo, Dec 16 '18 at 13:27

Mauro ALLEGRANZA · Accepted Answer · 2018-12-17T15:36:58.173

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.

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.

In language of the set should be equality symbol ? or defined equality symbol by $∈$ ? — Almot1960, Dec 17 '18 at 14:51

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

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