Inaccessible cardinal


Inaccessible cardinal

In set theory, an uncountable regular cardinal number is called weakly inaccessible if it is a weak limit cardinal, and strongly inaccessible, or just inaccessible, if it is a strong limit cardinal. Some authors do not require weakly and strongly inaccessible cardinals to be uncountable (in which case aleph_0 is strongly inaccessible).

Every strongly inaccessible cardinal is also weakly inaccessible, as every strong limit cardinal is also a weak limit cardinal. If the generalized continuum hypothesis holds, then a cardinal is strongly inaccessible if and only if it is weakly inaccessible.

aleph_0 (aleph-null) is a regular strong limit cardinal. Assuming the axiom of choice, every other infinite cardinal number is either regular or a (weak) limit. However, only a rather large cardinal number can be both and thus weakly inaccessible.

An ordinal is a weakly inaccessible cardinal if and only if it is a regular ordinal and it is a limit of regular ordinals. (Zero, one, and aleph_0 are regular ordinals, but not limits of regular ordinals.) A cardinal which is weakly inaccessible and also a strong limit cardinal is strongly inaccessible.

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.

Models and consistency

ZFC implies that the "V"κ is a model of ZFC whenever κ is strongly inaccessible. And ZF implies that the Gödel universe "L"κ is a model of ZFC whenever κ 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.

Suppose V is a model of ZFC. Either V contains no strong inaccessible or, taking κ to be the smallest strong inaccessible in V, "V"κ is a standard model of ZFC which contains no strong inaccessibles. So consistency of ZFC implies consistency of ZFC+"there are no strong inaccessibles". Similarly, either V contains no weak inaccessible or, taking κ to be the smallest ordinal which is weakly inaccessible relative to any standard sub-model of V, then "L"κ is a standard model of ZFC which contains no weak inaccessibles. So consistency of ZFC implies consistency of ZFC+"there are no weak inaccessibles".

If "V" is a standard model of ZFC and κ is an inaccessible in "V", then: "V"κ is one of the intended models of Zermelo–Fraenkel set theory; and Def ("V"κ) is one of the intended models of Von Neumann–Bernays–Gödel set theory; and "V"κ+1 is one of the intended models of Morse–Kelley set theory. Here Def ("X") is the Δ0 definable subsets of "X" (see constructible universe).

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 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). As is the case for the existence of any inaccessible cardinal, the inaccessible cardinal axiom is independent of 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 (which could 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

A cardinal κ is α-inaccessible, for α any ordinal, if and only if κ is inaccessible and for every ordinal β < α, the set of β-inaccessibles less than κ is unbounded in κ (and thus of cardinality κ, since κ is regular).

The α-inaccessible cardinals can be equivalently 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 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.

A cardinal κ is hyper-inaccessible if and only if κ is κ-inaccessible. (It can never be κ+1-inaccessible.)

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.

Using "weakly inaccessible" instead of "inaccessible", similar definitions can be made for "weakly α-inaccessible", "weakly hyper-inaccessible", and "weakly α-hyper-inaccessible".

See Mahlo cardinals.

ee also

*Club set
*Inner model
*Von Neumann universe
*Constructible universe

References

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