Talk:Universe

The set of all hereditary finite sets is a universe, but not a model of ZF (since ZF stipulates the axiom of infinity). 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. 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. Boris Tsirelson (talk) 21:39, 12 October 2017 (CEST)

Just an observation... If I am not mistaken, Gödel constructive sets 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 "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). Boris Tsirelson (talk) 22:18, 12 October 2017 (CEST)