# Differences

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set_universe [2015/08/25 22:49] nikolaj |
set_universe [2015/08/25 22:49] (current) nikolaj |
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In [[set theory]], the Tarski axiom states that there is a Grothendieck universe for every set. Note that this axiom implies the existence of strongly inaccessible cardinals and goes beyond ZFC. | In [[set theory]], the Tarski axiom states that there is a Grothendieck universe for every set. Note that this axiom implies the existence of strongly inaccessible cardinals and goes beyond ZFC. | ||

- | === Predicates === | + | == Motivation == |

- | | @#EEEE55: predicate | @#EEEE55: $X$... small set $\equiv X\in{\mathfrak U}_\mathrm{Sets}$ | | + | |

- | | + | |

- | === Motivation === | + | |

When one writes down a proposition in set theory, e.g. | When one writes down a proposition in set theory, e.g. | ||

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* Shielding from the famous set theoretical paradoxes | * Shielding from the famous set theoretical paradoxes | ||

* ${\mathfrak U}_\mathrm{Sets}$ in the object language, which easily lets one define a category of sets, [[Set]]. | * ${\mathfrak U}_\mathrm{Sets}$ in the object language, which easily lets one define a category of sets, [[Set]]. | ||

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+ | === Predicates === | ||

+ | | @#EEEE55: predicate | @#EEEE55: $X$... small set $\equiv X\in{\mathfrak U}_\mathrm{Sets}$ | | ||

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=== Requirements === | === Requirements === | ||

[[Grothendieck universe]], [[First infinite von Neumann ordinal]] | [[Grothendieck universe]], [[First infinite von Neumann ordinal]] |