grothendieck universe

In mathematics, if κ is a strongly inaccessible cardinal, the corresponding Grothendieck universe is the set of all sets with rank less than κ. By a theorem of Dimitry Mirimanoff, the Grothendieck universe is a set, not a proper class. Since it cannot be proven in ZFC that inaccessible cardinals exist, using a Grothendieck universe in the place of the class of all sets suffices for normal mathematical purposes, rendering proper classes unnecessary. The idea is due to Alexander Grothendieck, who used it as a way of avoiding proper classes in algebraic geometry. Grothendieck universes can be characterized by certain axiomatic properties, and it is possible to prove that any set satisfying such a property has cardinality equal to a strongly inaccessible cardinal. These properties are, in essence, what one needs to perform all the standard operations of set theory, and consequently it is possible to perform all of mathematics inside an appropriately chosen Grothendieck universe.

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