Category:Axioms/Axiom of Specification

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This category contains axioms related to Axiom of Specification.

For every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true.

Because we cannot quantify over functions, we need an axiom for every condition we can express.

Therefore, this axiom is sometimes called an axiom schema, as we introduce a lot of similar axioms.

This axiom schema can be formally stated as follows:

Set Theory

For any well-formed formula $\map P y$, we introduce the axiom:

$\forall z: \exists x: \forall y: \paren {y \in x \iff \paren {y \in z \land \map P y} }$

where each of $x$, $y$ and $z$ range over arbitrary sets.

Class Theory

The Axiom of Specification in the context of class theory has a similar form:

Let $\map \phi {A_1, A_2, \ldots, A_n, x}$ be a propositional function such that:

$A_1, A_2, \ldots, A_n$ are a finite number of free variables whose domain ranges over all classes
$x$ is a free variable whose domain ranges over all sets

Then the Axiom of Specification gives that:

$\forall A_1, A_2, \ldots, A_n: \exists B: \forall x: \paren {x \in B \iff \map \phi {A_1, A_2, \ldots, A_n, x} }$

where each of $B$ ranges over arbitrary classes.