Set

A set is a collection of different things, called elements or members, with the property that these elements may be compared by propositional equality. #m/def/set

“Unter einer Menge verstehen wir jede Zusammenfassung $𝑀$ von bestimmten wohlunterscheidbaren Objecten $𝑚$ unserer Anschauung oder unseres Denkens (welche die Elemente von $𝑀$ genannt werden) zu einem Ganze.”¹

In a material conception², two sets are said to be the same iff they have the same members, i.e.

(\forall 𝔐 𝐴, 𝔐 𝐵) [𝐴 = 𝐵 ⟺ (\forall 𝑥) [𝑥 \in 𝐴 ⟺ 𝑥 \in 𝐵]]

which is the Axiom of Extensionality. See axiomatic set theory for different axiomatic treatments of the set.

Further terms

It follows from extensionality that the Empty set $\emptyset$ is unique.
Subset

Forming sets

In a material conception

$𝐴 = {𝑎 1, 𝑎 2, \dots, 𝑎 𝑛}$ is the finite set with members $𝑎 1, 𝑎 2, \dots, 𝑎 𝑛$
$𝐴 = {𝑥 \in 𝐵 : 𝑃 (𝑥)}$ is the set of all $𝑥 \in 𝐵$ satisfying predicate $𝑃$ , i.e. $𝑥 \in 𝐴 ⟺ 𝑃 (𝑥)$
$𝐴 \cup 𝐵 = {𝑥 : 𝑥 \in 𝐴 \lor 𝑥 \in 𝐵}$ is the union of $𝐴$ and $𝐵$
$𝐴 \cap 𝐵 = {𝑥 : 𝑥 \in 𝐴 \land 𝑥 \in 𝐵}$ is the intersection of $𝐴$ and $𝐵$
$𝐴 ∖ 𝐵 = {𝑥 : 𝑥 \in 𝐴 \land 𝑥 \notin 𝐵}$ is the set difference of $𝐵$ from $𝐴$

In different foundations

The axiomatic set theories each give a notion of set.
In a Type theory with some notion of Equality, a set should be taken to be a type with Propositional equality.
- In particular, in (extensions of) ITT, a set should be taken to be an h-set.

#state/tidy | #lang/en | #SemBr

1895. Beiträge zur Begründung der transfiniten Mengenlehre. “By a set we understand any amalgamation $𝑀$ of definite, well distinguished objects $𝑚$ of our conception or our thought (which are called the elements of $𝑀$ ) to a [single] whole.” ↩
Such a statement becomes vacuous in a structural theory like ETCS, and outright wrong in Univalent Foundations. ↩

Set

Further terms

Forming sets

In different foundations

Footnotes