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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 $z\cup \{z\}.$

\[\exists X~(\emptyset \in X \wedge \forall~z(z\in X \Rightarrow z\cup \{z\}\in X).\]

| | | | | created: 2014-06-09 21:52:37 | modified: 2019-08-03 19:22:43 | by: bookofproofs | references: [656], [983]

1.Definition: Inductive Set

2.Definition: Minimal Inductive Set

3.Corollary: Minimal Inductive Set Is Subset Of All Inductive Sets

Edit or AddNotationAxiomatic Method

This work was contributed under CC BY-SA 4.0 by:

bookofproofs

This work is a derivative of:

CC BY-SA 4.0

Bibliography (further reading)

[983] Ebbinghaus, H.-D.: “Einführung in die Mengenlehre”, BI Wisschenschaftsverlag, 1994, 3

[656] Hoffmann, Dirk W.: “Grenzen der Mathematik - Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011