Editing Inaccessible cardinal (section)

==Existence of a proper class of inaccessibles==
There are many important axioms in set theory which assert the existence of a proper class of cardinals which satisfy a predicate of interest. In the case of inaccessibility, the corresponding axiom is the assertion that for every cardinal ''μ'', there is an inaccessible cardinal {{mvar|κ}} which is strictly larger, {{math|''μ'' < ''κ''}}. Thus, this axiom guarantees the existence of an infinite tower of inaccessible cardinals (and may occasionally be referred to as the inaccessible cardinal axiom). As is the case for the existence of any inaccessible cardinal, the inaccessible cardinal axiom is unprovable from the axioms of ZFC. Assuming ZFC, the inaccessible cardinal axiom is equivalent to the '''universe axiom''' of [[Grothendieck]] and [[Jean-Louis Verdier|Verdier]]: every set is contained in a [[Grothendieck universe]]. The axioms of ZFC along with the universe axiom (or equivalently the inaccessible cardinal axiom) are denoted ZFCU (not to be confused with ZFC with [[urelement]]s). This axiomatic system is useful to prove for example that every [[category (mathematics)|category]] has an appropriate [[Yoneda embedding]].

This is a relatively weak large cardinal axiom since it amounts to saying that ∞ is 1-inaccessible in the language of the next section, where ∞ denotes the least ordinal not in V, i.e. the class of all ordinals in your model.