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axiom of pairing (Axiom)

For any $ a$ and $ b$ there exists a set $ \{ a, b \}$ that contains exactly $ a$ and $ b$.

The Axiom of Pairing is one of the axioms of Zermelo-Fraenkel set theory. In symbols, it reads:

$\displaystyle \forall a \forall b \exists c \forall x (x \in c \leftrightarrow x = a \lor x = b). $
Using the Axiom of Extensionality, we see that the set $ c$ is unique, so it makes sense to define the pair
$\displaystyle \{ a, b \} =$    the unique $\displaystyle c$    such that $\displaystyle \forall x (x \in c \leftrightarrow x = a \lor x = b). $

Using the Axiom of Pairing, we may define, for any set $ a$, the singleton

$\displaystyle \{ a \} = \{ a, a \}. $

We may also define, for any set $ a$ and $ b$, the ordered pair

$\displaystyle (a, b) = \{ \{ a \}, \{ a, b \} \}. $

Note that this definition satisfies the condition

$\displaystyle (a, b) = (c, d)$    iff $\displaystyle a = c$    and $\displaystyle b = d. $

We may define the ordered $ n$-tuple recursively

$\displaystyle (a_1, \ldots, a_n) = ((a_1, \ldots, a_{n-1}), a_n). $



"axiom of pairing" is owned by Sabean.
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Other names:  pairing
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Cross-references: satisfies, ordered pair, singleton, axiom of extensionality, Zermelo-Fraenkel set theory, axioms, contains
There are 9 references to this entry.

This is version 4 of axiom of pairing, born on 2003-06-24, modified 2003-06-24.
Object id is 4392, canonical name is AxiomOfPairing.
Accessed 6656 times total.

Classification:
AMS MSC: 03E30 (Mathematical logic and foundations :: Set theory :: Axiomatics of classical set theory and its fragments)
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