Hereditarily finite set

From Apeirology Wiki
Revision as of 15:13, 31 August 2023 by RhubarbJayde (talk | contribs) (Created page with "A set is hereditarily finite if the smallest transitive set containing it is finite. So, an ordinal is hereditarily finite if and only if it is finite. Any hereditarily finite set is finite, but not every finite set is hereditarily finite, e.g. \(\{\omega\}\). One advantage of using hereditarily finite rather than finite sets is that they form a set, rather than a proper class. The set in question is \(V_\omega = L_\omega\) in the cumulative/Constructible hierarchy...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search

A set is hereditarily finite if the smallest transitive set containing it is finite. So, an ordinal is hereditarily finite if and only if it is finite. Any hereditarily finite set is finite, but not every finite set is hereditarily finite, e.g. \(\{\omega\}\). One advantage of using hereditarily finite rather than finite sets is that they form a set, rather than a proper class. The set in question is \(V_\omega = L_\omega\) in the cumulative/constructible hierarchies, while there is no \(\alpha\) so that \(V_\alpha\) or \(L_\alpha\) contains all the finite sets.

Retrieved from "https://apeirology.com/wiki/Hereditarily_finite_set?oldid=397"