Some authors do not require weakly and strongly inaccessible cardinals to be uncountable (in which case
Assuming the axiom of choice, every other infinite cardinal number is regular or a (weak) limit.
The assumption of the existence of a strongly inaccessible cardinal is sometimes applied in the form of the assumption that one can work inside a Grothendieck universe, the two ideas being intimately connected.
Zermelo–Fraenkel set theory with Choice (ZFC) implies that the
Thus, ZF together with "there exists a weakly inaccessible cardinal" implies that ZFC is consistent.
It is worth pointing out that the first claim can be weakened:
is a standard model of ZFC which contains no strong inaccessibles.
to be the smallest ordinal which is weakly inaccessible relative to any standard sub-model of
is a standard model of ZFC which contains no weak inaccessibles.
The issue whether ZFC is consistent with the existence of an inaccessible cardinal is more subtle.
There are arguments for the existence of inaccessible cardinals that cannot be formalized in ZFC.
One such argument, presented by Hrbáček & Jech (1999, p. 279), is that the class of all ordinals of a particular model M of set theory would itself be an inaccessible cardinal if there was a larger model of set theory extending M and preserving powerset of elements of M. 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 κ which is strictly larger, 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).
Assuming ZFC, the inaccessible cardinal axiom is equivalent to the universe axiom of Grothendieck and Verdier: every set is contained in a Grothendieck universe.
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.
The term "α-inaccessible cardinal" is ambiguous and different authors use inequivalent definitions.
The α-inaccessible cardinals can also be described as fixed points of functions which count the lower inaccessibles.
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 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 the definition that for any ordinal α, a cardinal κ is α-hyper-inaccessible if and only if κ is hyper-inaccessible and for every ordinal β < α, the set of β-hyper-inaccessibles less than κ is unbounded in κ. Hyper-hyper-inaccessible cardinals and so on can be defined in similar ways, and as usual this term is ambiguous.
(In fact, the set of such α is closed unbounded in κ.)
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.
Secondly, under ZFC Zermelo's categoricity theorem can be shown, which states that
is a standard model of (first order) ZFC.