fixed point of every ψβ for β < α (the value ψα(λ) is the λth such cardinal). This process of taking fixed points of functions generating successively larger cardinals is commonly encountered in the study of large cardinal numbers. The term hyper-inaccessible is ambiguous and has at least three incompatible meanings. Many authors use it to mean a regular limit of strongly inaccessible cardinals (1-inaccessible). Other authors use it to mean that is -inaccessible. (It can never be -inaccessible.) It is occasionally used to mean Mahlo cardinal. The term α-hyper-inaccessible is also ambiguous. Some authors use it to mean α-inaccessible. Other authors use the definition that for any ordinal α, a cardinal is α-hyper-inaccessible if and only if is hyper-inaccessible and for every ordinal β < α, the set of β-hyper-inaccessibles less than is unbounded in . Hyper-hyper-inaccessible cardinals and so on can be defined in similar ways, and as usual this term is ambiguous. Using "weakly inaccessible" instead of "inaccessible", similar definitions can be made for "weakly α-inaccessible", "weakly hyper-inaccessible", and "weakly α-hyper-inaccessible". Mahlo cardinals are inaccessible, hyper-inaccessible, hyper-hyper-inaccessible, ... and so on. Two model-theoretic characterisations of inaccessibility Firstly, a cardinal is inaccessible if and only if has the following reflection property: for all subsets , there exists such that is an elementary substructure of . (In fact, the set of such α is closed unbounded in .) Equivalently, is -indescribable for all n ≥ 0. It is provable in ZF that ∞ satisfies a somewhat weaker reflection property, where the substructure is only required to be 'elementary' with respect to a finite set of formulas. Ultimately, the reason for this weakening is that whereas the model-theoretic satisfaction relation can be defined, semantic truth itself (i.e. ) cannot, due to Tarski's theorem. Secondly, under ZFC it can be shown that is inaccessible if and only if is a model of second order ZFC. In this case, by the reflection property above, there exists such that is a standard model of (first order) ZFC. Hence, the existence of an inaccessible cardinal is a stronger hypothesis than the existence of a transitive model of ZFC. See also Worldly cardinal, a weaker notion Mahlo cardinal, a stronger notion Club set Inner model Von Neumann universe Constructible universe Works cited . English translation: . Large cardinals