TITLE: Is {Ø,{Ø},{Ø,{Ø}}, ... } the only known universe?
QUESTION [17 upvotes]: In the first pages of SGA4 I read

[...] Cependant le seul univers connu est l'ensemble des symboles du type {Ø,{Ø},{Ø,{Ø}}, ... } etc. (tous les éléments de cet univers sont des ensembles finis et cet univers est dénombrable). En particulier, on ne connaît pas d'univers qui contienne un élément de cardinal infini. [...]

(the sole known universe is like {Ø,{Ø},{Ø,{Ø}}, ... }, and we don't know any universe with a infinite cardinal).
Mais, c'est vrai? I wonder if during all these years somebody discovered a universe "bigger" than that exhibited by Grothendieck.

REPLY [17 votes]: Let me rephrase part of what Joel David Hamkins and Anon already said, but without mentioning inaccessible cardinals: 
A Grothendieck universe strictly bigger than the one in the question would be a model of ZFC.  (More precisely, it would become a model once we interpret the membership symbol of ZFC as actual membership.)  So the existence of such a Grothendieck universe would imply the consistency of ZFC.  G"oedel's second incompleteness theorem implies that ZFC cannot prove the consistency of ZFC.  Therefore ZFC cannot prove the existence of a Grothendieck universe.