Set properties

• unordered collection

• uniqueness of elements

• set comprehension

• well-definedness

Fundamental set properties

The most fundamental properties of a set are in terms of ordering and uniqueness: elements in a set are unordered (their order is immaterial), and the elements are unique (multiple instances of the same element are treated as a single object).

Sets can contain anything, but since we're concerned with math here, sets will contain well-defined mathematical objects. Being well-defined should enforce the requirement that only unambiguously defined candidates are to be accepted into a set.

A set is an unordered collection of distinct elements.

A set is uniquely determined by its elements, which is known as the uniqueness of elements, meaning that the only thing that defines or identifies a set are its elements.

Once defined a set is a standalone mathematical object.

The two most fundamental properties of sets is that they don't recognize order or repetition of elements.

The order in which the elements of a set are listed is immaterial, and so are the repetitions of elements. Therefore, these are equal:

$\{1,2,3\} = \{3,1,2\} = \{3,1,3,2,1,2\} = \{1,2,2,3,3\}$

The axiom of comprehension states that given any property, there is a set that contains all objects with that property. This position allows anything and everything to constitute a set.

Sets are allowed to contain anything, but by introducing the axiom of well-definedness, a set becomes a well-defined collection of objects that share a common property. Well-definedness states that there must be no ambiguity as to which elements belong to the set. However, this restriction alone cannot guarantee the consistency and clarity of what objects are exactly allowed to constitute a set.