Axiom:Axiom of Specification/Set Theory/Also presented as

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Axiom of Specification: Also presented as

The axiom of specification can also be specified as follows:

If $\phi$ is a property (with parameter $p$), then for any $X$ and $p$ there exists a set:
$Y = \paren {u \in X: \map \phi {u, p} }$
that contains all those $u \in X$ that have the property $\phi$.
$\forall X: \forall p: \exists Y: \forall u: \paren {u \in Y \iff \paren {u \in X \land \map \phi {u, p} } }$


Historical Note

The axiom of specification was created by Ernst Zermelo as a replacement for the comprehension principle of Frege set theory.

The latter had been demonstrated, via Russell's Paradox, to lead to the conclusion that Frege Set Theory is Logically Inconsistent.

Thus, rather than allowing a set to be constructed of any elements at all which satisfy a given property $P$, the elements in question are restricted to being elements of some pre-existing set.

This in turn leads to the further question of how to create such a pre-existing set in the first place.

Hence the need to develop further axioms in order to allow the creation of such sets.

As a result of this, Ernst Zermelo found it necessary to create:

the axiom of existence, allowing for the existence of $\O := \set {}$
the axiom of pairing, allowing for $\set {a, b}$ given the existence of $a$ and $b$
the axiom of unions, allowing for $\bigcup a$ given the existence of a set $a$ of sets
the axiom of powers, allowing for the power set $\powerset a$ to be generated for any set $a$
the axiom of infinity, allowing for the creation of the set of natural numbers $\N$.


Internationalization

Axiom of specification is translated:

In German: Aussorderungsaxiom  (literally: axiom of segregation)


Sources