Models and consistency
Zermelo–Fraenkel set theory with Choice (ZFC) implies that the th level of the Von Neumann universe is a model of ZFC whenever is strongly inaccessible. And ZF implies that the Gödel universe is a model of ZFC whenever is weakly inaccessible. Thus, ZF together with "there exists a weakly inaccessible cardinal" implies that ZFC is consistent. Therefore, inaccessible cardinals are a type of large cardinal. If is a standard model of ZFC and is an inaccessible in , then: is one of the intended models of Zermelo–Fraenkel set theory; and is one of the intended models of Mendelson's version of Von Neumann–Bernays–Gödel set theory which excludes global choice, replacing limitation of size by replacement and ordinary choice; and is one of the intended models of Morse–Kelley set theory. Here is the set of Δ0 definable subsets of ''X'' (see constructible universe). However, does not need to be inaccessible, or even a cardinal number, in order for to be a standard model of ZF (seeExistence 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 ''ψ''0(''λ'') the ''λ''th inaccessible cardinal, then the fixed points of ''ψ''0 are the 1-inaccessible cardinals. Then letting ''ψ''''β''(''λ'') be the ''λ''th ''β''-inaccessible cardinal, the fixed points of ''ψ''''β'' are the (''β''+1)-inaccessible cardinals (the values ''ψ''''β''+1(''λ'')). If ''α'' is a limit ordinal, an ''α''-inaccessible is a fixed point of every ''ψ''''β'' for ''β'' < ''α'' (the value ''ψ''''α''(''λ'') is the ''λ''th 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 (See also
* Worldly cardinal, a weaker notion * Mahlo cardinal, a stronger notion * Club set *Works cited
* * * * * *. English translation: . {{Mathematical logic Large cardinals