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set_universe [2015/08/25 22:48] nikolaj |
set_universe [2015/08/25 22:49] nikolaj |
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| === Discussion === | === Discussion === | ||
| A set universe ${\mathfrak U}_\mathrm{Sets}$ is a Grothendieck universe containing all sets generated by the [[first infinite von Neumann ordinal]] $\omega_{\mathcal N}$. | A set universe ${\mathfrak U}_\mathrm{Sets}$ is a Grothendieck universe containing all sets generated by the [[first infinite von Neumann ordinal]] $\omega_{\mathcal N}$. | ||
| - | + | It contains a model for the natural numbers, their powerset, the powersets of those etc. etc. I didn't specify //what's not// in such a universe, but for doing "normal non-foundational mathematics", one hardly ever needs anything that goes beyond a set obtained by a finite number of iterations of the applications of the power set operation on $\omega_{\mathcal N}$. | |
| - | A set universe contains a model for the natural numbers, their powerset, the powersets of those etc. etc. I didn't specify //what's not// in such a universe, but for doing "normal non-foundational mathematics", one hardly ever needs anything that goes beyond a set obtained by a finite number of iterations of the applications of the power set operation on $\omega_{\mathcal N}$. | + | |
| 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]]. | ||
| + | |||
| + | === Predicates === | ||
| + | | @#EEEE55: predicate | @#EEEE55: $X$... small set $\equiv X\in{\mathfrak U}_\mathrm{Sets}$ | | ||
| ----- | ----- | ||
| === Requirements === | === Requirements === | ||
| [[Grothendieck universe]], [[First infinite von Neumann ordinal]] | [[Grothendieck universe]], [[First infinite von Neumann ordinal]] | ||
