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 implies .
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 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 as Grenzzahlen.^{[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.
(alephnull) 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.
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MM70  Dima Sinapova  The tree property at the single and double successors of a singular cardinal
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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 Δ_{0} definable subsets of X (see constructible universe). However, does not need to be inaccessible, or even a cardinal number, in order for _{$\kappa$} to be a standard model of ZF (see below).
Suppose is a model of ZFC. Either V contains no strong inaccessible or, taking to be the smallest strong inaccessible in , 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 to be the smallest ordinal which is weakly inaccessible relative to any standard submodel of , then 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 nonexistence 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 1inaccessible 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.
αinaccessible cardinals and hyperinaccessible cardinals
The term "αinaccessible cardinal" is ambiguous and different authors use inequivalent definitions. One definition is that a cardinal κ is called αinaccessible, for any ordinal α, if κ is inaccessible and for every ordinal β < α, the set of βinaccessibles less than κ is unbounded in κ (and thus of cardinality κ, since κ is regular). In this case the 0inaccessible cardinals are the same as strongly inaccessible cardinals. Another possible definition is that a cardinal κ is called αweakly inaccessible if κ is regular and for every ordinal β < α, the set of βweakly inaccessibles less than κ is unbounded in κ. In this case the 0weakly inaccessible cardinals are the regular cardinals and the 1weakly inaccessible cardinals are the weakly inaccessible cardinals.
The αinaccessible 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 1inaccessible cardinals. Then letting ψ_{β}(λ) be the λ^{th} βinaccessible 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 αinaccessible 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 hyperinaccessible is ambiguous and has at least three incompatible meanings. Many authors use it to mean a regular limit of strongly inaccessible cardinals (1inaccessible). Other authors use it to mean that κ is κinaccessible. (It can never be κ+1inaccessible.) It is occasionally used to mean Mahlo cardinal.
The term αhyperinaccessible is also ambiguous. Some authors use it to mean αinaccessible. Other authors use the definition that for any ordinal α, a cardinal κ is αhyperinaccessible if and only if κ is hyperinaccessible and for every ordinal β < α, the set of βhyperinaccessibles less than κ is unbounded in κ.
Hyperhyperinaccessible 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 αinaccessible", "weakly hyperinaccessible", and "weakly αhyperinaccessible".
Mahlo cardinals are inaccessible, hyperinaccessible, hyperhyperinaccessible, ... and so on.
Two modeltheoretic characterisations of inaccessibility
Firstly, a cardinal κ is inaccessible if and only if κ 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 κ.) Therefore, 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 modeltheoretic 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 (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 is a property over .^{[2]}
See also
 Worldly cardinal, a weaker notion
 Mahlo cardinal, a stronger notion
 Club set
 Inner model
 Von Neumann universe
 Constructible universe
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 0444105352
 Hausdorff, Felix (1908), "Grundzüge einer Theorie der geordneten Mengen", Mathematische Annalen, 65 (4): 435–505, doi:10.1007/BF01451165, hdl:10338.dmlcz/100813, ISSN 00255831, S2CID 119648544
 Hrbáček, Karel; Jech, Thomas (1999), Introduction to set theory (3rd ed.), New York: Dekker, ISBN 9780824779153
 Kanamori, Akihiro (2003), The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nd ed.), Springer, ISBN 3540003843
 Sierpiński, Wacław; Tarski, Alfred (1930), "Sur une propriété caractéristique des nombres inaccessibles" (PDF), Fundamenta Mathematicae, 15: 292–300, doi:10.4064/fm151292300, ISSN 00162736
 Zermelo, Ernst (1930), "Über Grenzzahlen und Mengenbereiche: neue Untersuchungen über die Grundlagen der Mengenlehre" (PDF), Fundamenta Mathematicae, 16: 29–47, doi:10.4064/fm1612947, ISSN 00162736. 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 9780198532712.
References
 ^ A. Kanamori, "Zermelo and Set Theory", p.526. Bulletin of Symbolic Logic vol. 10, no. 4 (2004). Accessed 21 August 2023.
 ^ K. Hauser, "Indescribable cardinals and elementary embeddings". Journal of Symbolic Logic vol. 56, iss. 2 (1991), pp.439457.