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hereditarily_finite_set [2014/12/06 15:36] nikolaj |
hereditarily_finite_set [2015/10/08 13:43] nikolaj |
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| ===== Hereditarily finite set ===== | ===== Hereditarily finite set ===== | ||
| ==== Set ==== | ==== Set ==== | ||
| - | | @#FFBB00: definiendum | @#FFBB00: $V_\omega$ | | + | | @#FFBB00: definiendum | @#FFBB00: $V_\omega$ in it | |
| | @#55EE55: postulate | @#55EE55: $\emptyset\in V_\omega$ | | | @#55EE55: postulate | @#55EE55: $\emptyset\in V_\omega$ | | ||
| | @#FFFDDD: for all | @#FFFDDD: $x\in V_\omega$ | | | @#FFFDDD: for all | @#FFFDDD: $x\in V_\omega$ | | ||
| + | | @#55EE55: postulate | @#55EE55: ${\mathcal P}(x)\in V_\omega $ | | ||
| | @#55EE55: postulate | @#55EE55: $x = \emptyset\ \lor\ \exists (y\in V_\omega).\ x = {\mathcal P}(y) $ | | | @#55EE55: postulate | @#55EE55: $x = \emptyset\ \lor\ \exists (y\in V_\omega).\ x = {\mathcal P}(y) $ | | ||
| - | ==== Discussion ==== | + | ----- |
| + | === Discussion === | ||
| === Idea === | === Idea === | ||
| This is the set of all finite sets constructable when starting with $\emptyset$. It's the smallest infinite [[Grothendieck universe]], as well as a model of ZFC. | This is the set of all finite sets constructable when starting with $\emptyset$. It's the smallest infinite [[Grothendieck universe]], as well as a model of ZFC. | ||
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| === Reference === | === Reference === | ||
| Wikipedia: [[http://en.wikipedia.org/wiki/Hereditarily_finite_set|Hereditarily finite set]] | Wikipedia: [[http://en.wikipedia.org/wiki/Hereditarily_finite_set|Hereditarily finite set]] | ||
| - | ==== Parents ==== | + | |
| + | ----- | ||
| === Requirements === | === Requirements === | ||
| [[Power set]], [[Empty set]] | [[Power set]], [[Empty set]] | ||
