Axiom:Axioms of Class Existence
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Axiom
Let the ordered pair notation be understood to denote:
- $\tuple {X, Y} := \set {\set X, \set {X, Y}}$
that is, the Kuratowski formalization of ordered pairs.
The following are the axioms of class existence:
\((\text B 1)\) | $:$ | \(\ds \exists X: \forall u, v: \tuple {u, v} \in X \iff u \in v \) | $\in$-relation | ||||||
\((\text B 2)\) | $:$ | \(\ds \forall X, Y: \exists Z: \forall u: u \in Z \iff u \in X \land u \in Y \) | intersection | ||||||
\((\text B 3)\) | $:$ | \(\ds \forall X: \exists Z: \forall u: u \in Z \iff u \notin X \) | complement | ||||||
\((\text B 4)\) | $:$ | \(\ds \forall X: \exists Z: \forall u: u \in Z \iff \exists v: \tuple {u, v} \in X \) | domain | ||||||
\((\text B 5)\) | $:$ | \(\ds \forall X: \exists Z: \forall u, v: \tuple {u, v} \in Z \iff u \in X \) | |||||||
\((\text B 6)\) | $:$ | \(\ds \forall X: \exists Z: \forall u, v, w: \tuple {\tuple {u, v}, w} \in Z \iff \tuple {\tuple {v, w}, u} \in X \) | |||||||
\((\text B 7)\) | $:$ | \(\ds \forall X: \exists Z: \forall u, v, w: \tuple {\tuple {u, v}, w} \in Z \iff \tuple {\tuple {u, w}, v} \in X \) |
Sources
- 2010: Elliott Mendelson: Introduction to Mathematical Logic (5th ed.): $4$ Axiomatic Set Theory