Axiom of infinity: Difference between revisions
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RhubarbJayde (talk | contribs) (Created page with "The axiom of infinity is a common mathematical axiom included in theories such as Kripke-Platek set theory or ZFC. It asserts that there exists an inductive set - i.e. a set \(x\) so that \(0 \in x\) and, if \(n \in x\), then \(n+1 \in x\). By using \(\Delta_0\)-separation, this implies that \(\omega\) exists. The axiom of infinity, obviously, drastically increases the strength of set theory, since else one is not at all able to define Ordinal|ordinal...") |
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The axiom of infinity is a common mathematical axiom included in theories such as [[Kripke-Platek set theory]] or [[ZFC]]. It asserts that there exists an inductive set - i.e. a set \(x\) so that \(0 \in x\) and, if \(n \in x\), then \(n+1 \in x\). By using \(\Delta_0\)-separation, this implies that [[Omega|\(\omega\)]] exists. The axiom of infinity, obviously, drastically increases the strength of set theory, since else one is not at all able to define [[Ordinal|ordinals]]. For example, \(V_\omega\), the set of [[Hereditarily finite set|hereditarily finite sets]], is a model of [[ZFC]] minus the axiom of infinity.
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