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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.
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