Inaccessible cardinal

In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal $κ$ is strongly inaccessible if it satisfies the following three conditions: it is uncountable, it is not a sum of fewer than $κ$ cardinals smaller than $κ$ , and $\alpha <\kappa$ implies $2^{\alpha }<\kappa$ .

The term "inaccessible cardinal" is ambiguous. Until about 1950, it meant "weakly inaccessible cardinal", but since then it usually means "strongly inaccessible cardinal". An uncountable cardinal is weakly inaccessible if it is a regular weak limit cardinal. It is strongly inaccessible, or just inaccessible, if it is a regular strong limit cardinal (this is equivalent to the definition given above). 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); in the latter they were referred to along with $\aleph _{0}$ as Grenzzahlen (English "limit numbers").^[1]

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 $ω$ 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

Suppose that $\kappa$ is a cardinal number. Zermelo–Fraenkel set theory with Choice (ZFC) implies that the $\kappa$ th level of the Von Neumann universe $V_{\kappa }$ is a model of ZFC whenever $\kappa$ is strongly inaccessible. Furthermore, ZF implies that the Gödel universe $L_{\kappa }$ is a model of ZFC whenever $\kappa$ 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 $\kappa$ is an inaccessible in $V$ , then

$V_{\kappa }$ is one of the intended models of Zermelo–Fraenkel set theory;
$Def(V_{\kappa })$ 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 $V_{\kappa +1}$ is one of the intended models of Morse–Kelley set theory.

Here, $Def(X)$ is the set of Δ₀-definable subsets of X (see constructible universe). It is worth pointing out that the first claim can be weakened: $\kappa$ does not need to be inaccessible, or even a cardinal number, in order for $V$ to be a standard model of ZF (see below).

Suppose $V$ is a model of ZFC. Either $V$ contains no strong inaccessible or, taking $\kappa$ to be the smallest strong inaccessible in $V$ , $V_{\kappa }$ 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 $\kappa$ to be the smallest ordinal which is weakly inaccessible relative to any standard sub-model of $V$ , then $L_{\kappa }$ 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 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.

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 (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 $κ$ is called α-inaccessible, for any ordinal α, 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 $U\subset V_{\kappa }$ , there exists $\alpha <\kappa$ 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 $κ$ .) Therefore, $\kappa$ is $\Pi _{n}^{0}$ -indescribable for all n ≥ 0. On the other hand, there is not necessarily an ordinal $\alpha >\kappa$ such that $V_{\kappa }$ , and if this holds, then $\kappa$ must be the $\kappa$ th inaccessible cardinal.^[2]

It is provable in ZF that $V$ has a somewhat weaker reflection property, where the substructure $(V_{\alpha },\in ,U\cap V_{\alpha })$ 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. $\vDash _{V}$ ) cannot, due to Tarski's theorem.

Secondly, under ZFC Zermelo's categoricity theorem can be shown, which states that $\kappa$ is inaccessible if and only if $(V_{\kappa },\in )$ is a model of second order ZFC.

In this case, by the reflection property above, there exists $\alpha <\kappa$ such that $(V_{\alpha },\in )$ 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.

Inaccessibility of $\kappa$ is a $\Pi _{1}^{1}$ property over $V_{\kappa }$ ,^[3] while a cardinal $\pi$ being inaccessible (in some given model of $\mathrm {ZF}$ containing $\pi$ ) is $\Pi _{1}$ .^[4]

Works cited

Drake, F. R. (1974), Set Theory: An Introduction to Large Cardinals, Studies in Logic and the Foundations of Mathematics, vol. 76, Elsevier Science, ISBN 0-444-10535-2
Hausdorff, Felix (1908), "Grundzüge einer Theorie der geordneten Mengen", Mathematische Annalen, 65 (4): 435–505, doi:10.1007/BF01451165, hdl:10338.dmlcz/100813, ISSN 0025-5831, S2CID 119648544
Hrbáček, 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, doi:10.4064/fm-15-1-292-300, ISSN 0016-2736
Zermelo, Ernst (1930), "Über Grenzzahlen und Mengenbereiche: neue Untersuchungen über die Grundlagen der Mengenlehre" (PDF), Fundamenta Mathematicae, 16: 29–47, doi:10.4064/fm-16-1-29-47, ISSN 0016-2736. English translation: Ewald, William B. (1996), "On boundary numbers and domains of sets: new investigations in the foundations of set theory", From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, pp. 1208–1233, ISBN 978-0-19-853271-2.

References

^ A. Kanamori, "Zermelo and Set Theory", p.526. Bulletin of Symbolic Logic vol. 10, no. 4 (2004). Accessed 21 August 2023.
^ A. Enayat, "Analogues of the MacDowell-Specker_theorem for set theory" (2020), p.10. Accessed 9 March 2024.
^ K. Hauser, "Indescribable cardinals and elementary embeddings". Journal of Symbolic Logic vol. 56, iss. 2 (1991), pp.439--457.
^ K. J. Devlin, "Indescribability Properties and Small Large Cardinals" (1974). In $\vDash$ ISILC Logic Conference: Proceedings of the International Summer Institute and Logic Colloquium, Kiel 1974, Lecture Notes in Mathematics, vol. 499 (1974)

[1] A. Kanamori, "Zermelo and Set Theory", p.526. Bulletin of Symbolic Logic vol. 10, no. 4 (2004). Accessed 21 August 2023.

[2] A. Enayat, "Analogues of the MacDowell-Specker_theorem for set theory" (2020), p.10. Accessed 9 March 2024.

[3] K. Hauser, "Indescribable cardinals and elementary embeddings". Journal of Symbolic Logic vol. 56, iss. 2 (1991), pp.439--457.

[4] K. J. Devlin, "Indescribability Properties and Small Large Cardinals" (1974). In $\vDash$ ISILC Logic Conference: Proceedings of the International Summer Institute and Logic Colloquium, Kiel 1974, Lecture Notes in Mathematics, vol. 499 (1974)

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