Axiom:Axiom of Unions/Set Theory
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Axiom
For every set of sets $A$, there exists a set $x$ (the union set) that contains all and only those elements that belong to at least one of the sets in the $A$:
- $\forall A: \exists x: \forall y: \paren {y \in x \iff \exists z: \paren {z \in A \land y \in z} }$
Also known as
The axiom of unions is in fact most frequently found with the name axiom of union.
However, in some treatments of axiomatic set theory and class theory, for example Morse-Kelley set theory this name is used to mean something different.
Hence $\mathsf{Pr} \infty \mathsf{fWiki}$ specifically uses the plural form axiom of unions for this, and reserves the singular form axiom of union for that.
Other terms that can be found to refer to the axiom of unions:
- the axiom of the sum set
- the axiom of amalgamation
- the union axiom.
Also see
- Results about the axiom of unions can be found here.
Sources
- 1955: John L. Kelley: General Topology: Appendix: Functions: Axiom $\text{VI}$
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 4$: Unions and Intersections
- 1982: Alan G. Hamilton: Numbers, Sets and Axioms ... (previous) ... (next): $\S 4$: Set Theory: $4.2$ The Zermelo-Fraenkel axioms: $\text {ZF4}$
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 5$ The union axiom: Note $1$
- Weisstein, Eric W. "Zermelo-Fraenkel Axioms." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Zermelo-FraenkelAxioms.html
- Weisstein, Eric W. "Axiom of the Sum Set." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AxiomoftheSumSet.html