nLab pairing structure

Definition

Pairing structure

Given a set $V$ with an extensional relation $\prec$ on $V$, a pairing structure is a binary function $P:V \times V \to V$ such that

• for all $a \in V$ and $b \in V$, $a \prec P(a, b)$ and $b \prec P(a, b)$

Unordered pairing structure

An unordered pairing structure on $V$ is pairing structure $\{-,-\}:V \times V \to V$ where additionally

• for all sets $z$, $z \in \{x, y\}$ implies that $z = x$ or $z = y$.

Uniqueness of $\{x, y\}$ follows from $\prec$ being an extensional relation.

Ordered pairing structure

An ordered pairing structure on $V$ is a pairing structure $(-,-):V \times V \to V$ which satisfies product extensionality:

• for all $a \in V$, $a' \in V$, $b \in V$, $b' \in V$, $P(a, b) = P(a', b')$ if and only if $a = a'$ and $b = b'$

Foundational concerns

In any material set theory, instead of postulating the mere existence of a set $P$ in which $a \in P$ and $b \in P$ one could add a primitive binary operation $P(a, b)$ which takes material sets $a$ and $b$ and returns a material set $P(a, b)$ such that for all $a$ and $b$, $a \in P(a, b)$ and $b \in P(a, b)$

See also

• axiom of pairing

• union structure

• powerset structure

• natural numbers structure

• product extensionality

References

• Håkon Robbestad Gylterud, Elisabeth Stenholm, Niccolò Veltri Terminal Coalgebras and Non-wellfounded Sets in Homotopy Type Theory [arXiv:2001.06696]