Set

Definition 1.

Naively, a set is a collection of objects. There are two foundational relationships between sets: One is equality $=$. The other one is membership $\in$: For a set $A$ and an object $x$, one writes

\[x \in A\]

to mean “$x$ is a member of $A$” or “$x$ belongs to $A$”.

Formally, if we work in $\ZF$, any discussed object is a set. $\in$ is merely a symbol used to give structures to sets.

The meaning of sets is purely shaped by axioms. To understand sets is to understand the axioms. Just as the famous quote from Ludiwg Wittgenstein goes¹:

… the meaning of a word is its use in the langugage.