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 which is strictly larger, . 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 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 urelements). This axiomatic system is useful to prove for example that every 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. α-inaccessible cardinals and hyper-inaccessible cardinals The term "α-inaccessible cardinal" is ambiguous and different authors use inequivalent definitions. One definition is that a cardinal is called α-inaccessible, for α any ordinal, if is inaccessible and for every ordinal β < α, the set of β-inaccessibles less than is unbounded in (and thus of cardinality , since is regular). In this case the 0-inaccessible cardinals are the same as strongly inaccessible cardinals. Another possible definition is that a cardinal is called α-weakly inaccessible if is regular and for every ordinal β < α, the set of β-weakly inaccessibles less than is unbounded in κ. In this case the 0-weakly inaccessible cardinals are the regular cardinals and the 1-weakly inaccessible cardinals are the weakly inaccessible cardinals. The α-inaccessible cardinals can also be described as fixed points of functions which count the lower inaccessibles. For example, denote by ψ0(λ) the λth inaccessible cardinal, then the fixed points of ψ0 are the 1-inaccessible cardinals. Then letting ψβ(λ) be the λth β-inaccessible cardinal, the fixed points of ψβ are the (β+1)-inaccessible cardinals (the values ψβ+1(λ)). If α is a limit ordinal, an α-inaccessible is a