The Kuratowski ordered pair $(a,b) := \{\{a\},\{a,b\}\}$

For sets $a$ and $b$, the ordered pair $(a,b)$ is the set

$(a,b) := \{\{a\},\{a,b\}\},$

formed from the unordered pairs and singletons of The unordered pair $\{x,y\}$ and the singleton $\{x\} = \{x,x\}$; three applications of The Axiom of Pairing: $\forall x\,\forall y\,\exists z\,\forall t\,(t \in z \leftrightarrow (t = x \vee t = y))$ produce it. The first coordinate is $a$ and the second is $b$.

When $a = b$ the two members coincide, since $\{a,b\} = \{a,a\} = \{a\}$, and the pair degenerates to $(a,a) = \{\{a\}\}$.

• Why this set and not another. An ordered pair is required to satisfy one property, that $(a,b) = (c,d)$ exactly when $a = c$ and $b = d$; that is $(a,b) = (c,d)$ if and only if $a = c$ and $b = d$, and it is the only thing any later construction uses. Other definitions with the same property exist, and nothing below distinguishes them from this one.

• The degenerate case is where a careless proof fails. An argument that treats $\{\{a\},\{a,b\}\}$ as a set with two distinct members breaks at $a = b$, and that case has to be handled separately in the proof of the characterising property.