Ordered Pair/Kuratowski Formalization/Motivation

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The only reason for the Kuratowski formalization of ordered pairs:

$\tuple {a, b} = \set {\set a, \set {a, b} }$

is so their existence can be justified in the strictures of the axiomatic set theory, in particular Zermelo-Fraenkel set theory. Once that has been demonstrated, there is no need to invoke it again.

The fact that this formulation allows that:

$\tuple {a, b} = \tuple {c, d} \iff a = c, b = d$

is its stated aim.

The fact that $\set {a, b} \in \tuple {a, b}$ is an unfortunate side-effect brought about by means of the definition.

It would be possible to add another axiom to ZF or ZFC specifically to allow for ordered pairs to be defined, and in some systems of axiomatic set theory this is what is done.