All of the axioms introduce so far do not ensure the existence of infinite sets. The following axiom closes this gap.
Axiom: Axiom of Infinity
There exists a set \(X\) containing the empty set and also with every element $z$ also the element $z\cup \{z\}.$
\[\exists X~(\emptyset \in X \wedge \forall~z(z\in X \Rightarrow z\cup \{z\}\in X).\]
![axiominfinity](https://github.com/bookofproofs/bookofproofs.github.io/blob/main/_sources/_assets/images/examples/axiominfinity.png?raw=true)
Table of Contents
Corollaries: 1
- Definition: Inductive Set
- Definition: Minimal Inductive Set
Mentioned in:
Axioms: 1
Definitions: 2 3 4 5
Examples: 6
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References
Bibliography
- Hoffmann, Dirk W.: "Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik", Spektrum Akademischer Verlag, 2011
- Ebbinghaus, H.-D.: "Einführung in die Mengenlehre", BI Wisschenschaftsverlag, 1994, 3th Edition