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. |
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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Revision as of 05:58, 25 March 2024
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 ordinals. For example, \(V_\omega\), the set of hereditarily finite sets, is a model of ZFC minus the axiom of infinity.