View source for Talk:Universe

The set of all [https://en.wikipedia.org/wiki/Hereditarily_finite_set hereditary finite sets] is a universe, but not a model of ZF (since ZF stipulates [[Infinity, axiom of|the axiom of infinity]]). [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 20:33, 12 October 2017 (CEST)
:That does indeed appear to be the case.  There is a definitional choice: (0) allow the empty set to be a universe; (1) require a universe to have an element (equivalently to have the empty set as an element); (2) require a universe to have an infinite set as an element (such as the natural numbers).  Allowing the hereditarily finite sets to be a universe makes $\aleph_0$ the first inaccessible cardinal.  [[User:Richard Pinch|Richard Pinch]] ([[User talk:Richard Pinch|talk]]) 20:58, 12 October 2017 (CEST)
::Yes. On Wikipedia, only uncountable cardinals are classified into accessible and inaccessible. I have no appropriate books on my shell now, thus I do not know, whether that is the consensus, or not. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 21:39, 12 October 2017 (CEST)
:::One book found: T. J. Jech "Lectures in set theory with particular emphasis on the method of forcing" Springer 1971. There, only uncountable cardinals may be called strongly inaccessible. Universe probably does not appear there. [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 22:41, 12 October 2017 (CEST)

Just an observation... If I am not mistaken, [[Gödel constructive set]]s provide an example of a transitive model of ZF (even ZFC) but (possibly) not a universe. It happens because the first axiom of universe accepts ''arbitrary'' families. And nevertheless, on Wikipedia I see "[https://en.wikipedia.org/wiki/Constructible_universe Gödel's constructible universe]". I guess, "universe" is a rather fuzzy idea, without a consensus about the definition (and on WP it is treated as fuzzy). [[User:Boris Tsirelson|Boris Tsirelson]] ([[User talk:Boris Tsirelson|talk]]) 22:18, 12 October 2017 (CEST)