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). Weakly inaccessible cardinals were introduced by Hausdorff (1908), and strongly inaccessible ones by Sierpiński & Tarski (1930) and Zermelo (1930).

The term "inaccessible cardinal" is ambiguous. Until about 1950 it meant "weakly inaccessible cardinal", but since then it usually means "strongly inaccessible cardinal".

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 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.

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). However, κ does not need to be inaccessible, or even a cardinal number, in order for V_κ to be a standard model of ZF (see below).

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. Thus, the 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". This shows that ZFC cannot prove the existence of an inaccessible cardinal, so ZFC is consistent with the non-existence of any inaccessible cardinals.

The issue whether ZFC is consistent with the existence of an inaccessible cardinal is more subtle. The proof sketched in the previous paragraph that the consistency of ZFC implies the consistency of ZFC + "there is not an inaccessible cardinal" can be formalized in ZFC. However, assuming that ZFC is consistent, no proof that the consistency of ZFC implies the consistency of ZFC + "there is an inaccessible cardinal" can be formalized in ZFC. This follows from Gödel's second incompleteness theorem, which shows that if ZFC + "there is an inaccessible cardinal" is consistent, then it cannot prove its own consistency. Because ZFC + "there is an inaccessible cardinal" does prove the consistency of ZFC, if ZFC proved that its own consistency implies the consistency of ZFC + "there is an inaccessible cardinal" then this latter theory would be able to prove its own consistency, which is impossible if it is consistent.

There are arguments for the existence of inaccessible cardinals that cannot be formalized in ZFC. One such argument, presented by Hrbacek & 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.

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 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 (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

The term "α-inaccessible cardinal" is ambiguous and different authors use inequivalent definitions. One definition is that a cardinal κ is called α-inaccessible, for α any ordinal, 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). In this case the 0-inaccessible cardinals are the same as strongly inaccessible cardinals. Another possible definition is that a cardinal κ is called α-weakly inaccessible if κ is regular and for every ordinal β < α, the set of β-weakly inaccessibles less than κ 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 λ^th inaccessible cardinal, then the fixed points of ψ₀ 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 κ is κ-inaccessible. (It can never be κ+1-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 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.

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 κ is inaccessible if and only if κ has the following reflection property: for all subsets U ⊂ V_κ, there exists α < κ such that $(V_\alpha,\in,U\cap V_\alpha)$ is an elementary substructure of $(V_\kappa,\in,U)$ . (In fact, the set of such α is closed unbounded in κ.) Equivalently, κ is $\Pi_n^0$ -indescribable for all n ≥ 0.

It is provable in ZF that ∞ satisfies a somewhat weaker reflection property, where the substructure (V_α, ∈, U ∩ V_α) 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 $\models$ can be defined, truth itself cannot, due to Tarski's theorem.

Secondly, under ZFC it can be shown that κ is inaccessible if and only if (V_κ, ∈) is a model of second order ZFC.

In this case, by the reflection property above, there exists α < κ such that (V_α, ∈) is a standard model of (first order) ZFC. Hence, the existence of an inaccessible cardinal is a stronger hypothesis than the existence of a standard model of ZFC.

References

Drake, F. R. (1974), Set Theory: An Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics 76, Elsevier Science Ltd, ISBN 0-444-10535-2
Hausdorff, Felix (1908), "Grundzüge einer Theorie der geordneten Mengen", Mathematische Annalen 65 (4): 435–505, doi:10.1007/BF01451165, ISSN 0025-5831
Hrbacek, Karel; Jech, Thomas (1999), Introduction to set theory (3rd ed.), New York: Dekker, ISBN 978-0-8247-7915-3
Kanamori, Akihiro (2003), The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nd ed.), Springer, ISBN 3-540-00384-3
Sierpiński, Wacław; Tarski, Alfred (1930), "Sur une propriété caractéristique des nombres inaccessibles" (PDF), Fundamenta Mathematicae 15: 292–300, ISSN 0016-2736
Zermelo, Ernst (1930), "Über Grenzzablen und Mengenbereiche", Fundamenta Mathematicae 16: 29–47, ISSN 0016-2736

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