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set_universe [2014/12/07 22:00] nikolaj |
set_universe [2015/08/25 22:48] nikolaj |
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| | @#55EE55: postulate | @#55EE55: $\omega_{\mathcal N}\subseteq {\mathfrak U}_\mathrm{Sets}$ | | | @#55EE55: postulate | @#55EE55: $\omega_{\mathcal N}\subseteq {\mathfrak U}_\mathrm{Sets}$ | | ||
| - | ==== Discussion ==== | + | ----- |
| - | === Idea === | + | === 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]] $\omega_{\mathcal N}$. | ||
| - | === Elaboration === | ||
| 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 $\omega_{\mathcal N}$. | 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 $\omega_{\mathcal N}$. | ||
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| * ${\mathfrak U}_\mathrm{Sets}$ 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]]. | ||
| - | ==== Parents ==== | + | ----- |
| === Requirements === | === Requirements === | ||
| [[Grothendieck universe]], [[First infinite von Neumann ordinal]] | [[Grothendieck universe]], [[First infinite von Neumann ordinal]] | ||
