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set_universe [2015/08/25 22:47] nikolaj |
set_universe [2015/08/25 22:49] nikolaj |
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| === Discussion === | === Discussion === | ||
| - | == Idea == | ||
| 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 $\omega_{\mathcal N}$. | |
| - | === 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 $\omega_{\mathcal N}$. | + | |
| 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. | ||
