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Difference between revisions of "Talk:Universe"

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(Undo revision 42058 by Boris Tsirelson (talk) or not?)
(fuzzy idea)
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: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)
 
: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)
 
::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)
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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)

Revision as of 20:18, 12 October 2017

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)

How to Cite This Entry:
Universe. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Universe&oldid=42060