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Inset theory, acardinal number is astrongly inaccessible cardinal if it isuncountable,regular, and astrong limit cardinal.A cardinal is aweakly inaccessible cardinal if it is uncountable, regular, and aweak limit cardinal.

Since about 1950, "inaccessible cardinal" has typically meant "strongly inaccessible cardinal" whereas before it had meant "weakly inaccessible cardinal". Weakly inaccessible cardinals were introduced byHausdorff (1908). Strongly inaccessible cardinals were introduced bySierpiński & Tarski (1930) andZermelo (1930); in the latter they were referred to along with${\displaystyle {\mathrm{\aleph }}_0}$ asGrenzzahlen (English "limit numbers").^[1]

Every strongly inaccessible cardinal is a weakly inaccessible cardinal. Thegeneralized continuum hypothesis implies that all weakly inaccessible cardinals are strongly inaccessible as well.

The two notions of an inaccessible cardinal${\displaystyle \kappa }$ describe a cardinality${\displaystyle \kappa }$ which can not be obtained as the cardinality of a result of typical set-theoretic operations involving only sets of cardinality less than${\displaystyle \kappa }$. Hence the word "inaccessible". By mandating that inaccessible cardinals are uncountable, they turn out to be very large.

In particular, inaccessible cardinals need not exist at all. That is, it is believed that there are models ofZermelo-Fraenkel set theory, even with theaxiom of choice (ZFC), for which no inaccessible cardinals exist^[2]. On the other hand, it also believed that there are models of ZFC for which even strongly inaccessible cardinalsdo exist. That ZFC can accommodate these large sets, but does not necessitate them, provides an introduction to thelarge cardinal axioms. See alsoModels and consistency.

The existence of a strongly inaccessible cardinal is equivalent to the existence of aGrothendieck universe.If${\displaystyle \kappa }$ is a strongly inaccessible cardinal then thevon Neumann stage${\displaystyle V_{\kappa }}$ is a Grothendieck universe.Conversely, if${\displaystyle U}$ is a Grothendieck universe then there is a strongly inaccessible cardinal${\displaystyle \kappa }$ such that${\displaystyle V_{\kappa }=U}$. As expected from their correspondence with strongly inaccessible cardinals, Grothendieck universes are very well-closed under set-theoretic operations.

Anordinal is a weakly inaccessible cardinal if and only if it is a regular ordinal and it is a limit of regular ordinals. (Zero, one, andω are regular ordinals, but not limits of regular ordinals.)

From some perspectives, the requirement that a weakly or strongly inaccessible cardinal be uncountable is unnatural or unnecessary. Even though⁠${\displaystyle {\mathrm{\aleph }}_0}$⁠ is countable, it is regular and is a strong limit cardinal.⁠${\displaystyle {\mathrm{\aleph }}_0}$⁠ is also the smallest weak limit regular cardinal. Assuming the axiom of choice, every other infinite cardinal number is either regular or a weak limit cardinal. However, only a rather large cardinal number can be both. Since a cardinal⁠${\displaystyle \kappa }$⁠ larger than⁠${\displaystyle {\mathrm{\aleph }}_0}$⁠ is necessarily uncountable, if⁠${\displaystyle \kappa }$⁠ is also regular and a weak limit cardinal then⁠${\displaystyle \kappa }$⁠ must be a weakly inaccessible cardinal.

Suppose that${\displaystyle \kappa }$ is a cardinal number.Zermelo–Fraenkel set theory with Choice (ZFC) implies that the${\displaystyle \kappa }$th level of theVon Neumann universe${\displaystyle V_{\kappa }}$ is amodel of ZFC whenever${\displaystyle \kappa }$ is strongly inaccessible. Furthermore, ZF implies that theGödel universe${\displaystyle L_{\kappa }}$ is a model of ZFC whenever${\displaystyle \kappa }$ is weakly inaccessible. Thus, ZF together with "there exists a weakly inaccessible cardinal" implies that ZFC is consistent. Therefore, inaccessible cardinals are a type oflarge cardinal.

If${\displaystyle V}$ is a standard model of ZFC and${\displaystyle \kappa }$ is an inaccessible in${\displaystyle V}$, then

1. ${\displaystyle V_{\kappa }}$ is one of the intended models ofZermelo–Fraenkel set theory;
2. ${\displaystyle Def(V_{\kappa })}$ is one of the intended models of Mendelson's version ofVon Neumann–Bernays–Gödel set theory which excludes global choice, replacing limitation of size by replacement and ordinary choice;
3. and${\displaystyle V_{\kappa +1}}$ is one of the intended models ofMorse–Kelley set theory.

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

Suppose${\displaystyle V}$ is a model of ZFC. Either${\displaystyle V}$ contains no strong inaccessible or, taking${\displaystyle \kappa }$ to be the smallest strong inaccessible in${\displaystyle V}$,${\displaystyle V_{\kappa }}$ 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, eitherV contains no weak inaccessible or, taking${\displaystyle \kappa }$ to be the smallest ordinal which is weakly inaccessible relative to any standard sub-model of${\displaystyle V}$, then${\displaystyle L_{\kappa }}$ 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 fromGödel'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 byHrbáček & Jech (1999, p. 279), is that the class of all ordinals of a particular modelM of set theory would itself be an inaccessible cardinal if there was a larger model of set theory extendingM and preserving powerset of elements ofM.

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 cardinalμ, there is an inaccessible cardinalκ 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 theuniverse axiom ofGrothendieck andVerdier: every set is contained in aGrothendieck 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 withurelements). This axiomatic system is useful to prove for example that everycategory has an appropriateYoneda 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. One definition is thata cardinalκ is calledα-inaccessible, for any ordinalα, ifκ is inaccessible and for every ordinalβ <α, the set ofβ-inaccessibles less thanκ is unbounded inκ (and thus of cardinalityκ, sinceκ is regular). In this case the 0-inaccessible cardinals are the same as strongly inaccessible cardinals. Another possible definition is that a cardinalκ is calledα-weakly inaccessible ifκ is regular and for every ordinalβ <α, the set ofβ-weakly inaccessibles less thanκ 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.

Theα-inaccessible cardinals can also be described as fixed points of functions which count the lower inaccessibles. For example, denote byψ₀(λ) theλ^th inaccessible cardinal, then the fixed points ofψ₀ are the 1-inaccessible cardinals. Then lettingψ_β(λ) be theλ^thβ-inaccessible cardinal, the fixed points ofψ_β are the (β+1)-inaccessible cardinals (the valuesψ_β+1(λ)). Ifα is a limit ordinal, anα-inaccessible is a fixed point of everyψ_β forβ <α (the valueψ_α(λ) is theλ^th such cardinal). This process of taking fixed points of functions generating successively larger cardinals is commonly encountered in the study oflarge cardinal numbers.

The termhyper-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 thatκ isκ-inaccessible. (It can never beκ+1-inaccessible.) It is occasionally used to meanMahlo cardinal.

The termα-hyper-inaccessible is also ambiguous. Some authors use it to meanα-inaccessible. Other authors use the definition thatfor any ordinalα, a cardinalκ isα-hyper-inaccessible if and only ifκ is hyper-inaccessible and for every ordinalβ <α, the set ofβ-hyper-inaccessibles less thanκ is unbounded inκ.

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 "weaklyα-inaccessible", "weakly hyper-inaccessible", and "weaklyα-hyper-inaccessible".

Mahlo cardinals are inaccessible, hyper-inaccessible, hyper-hyper-inaccessible, ... and so on.

Firstly, a cardinalκ is inaccessible if and only ifκ has the followingreflection property: for all subsets${\displaystyle U\subset V_{\kappa }}$, there exists such that${\displaystyle (V_{\alpha },\in ,U\cap V_{\alpha })}$ is anelementary substructure of${\displaystyle (V_{\kappa },\in ,U)}$. (In fact, the set of suchα isclosed unbounded inκ.) Therefore,${\displaystyle \kappa }$ is${\displaystyle {\mathrm{\Pi }}_n^0}$-indescribable for alln ≥ 0. On the other hand, there is not necessarily an ordinal${\displaystyle \alpha >\kappa }$ such that${\displaystyle V_{\kappa }}$, and if this holds, then${\displaystyle \kappa }$ must be the${\displaystyle \kappa }$th inaccessible cardinal.^[3]

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

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

In this case, by the reflection property above, there exists such that${\displaystyle (V_{\alpha },\in )}$ 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${\displaystyle \kappa }$ is a${\displaystyle {\mathrm{\Pi }}_1^1}$ property over${\displaystyle V_{\kappa }}$,^[4] while a cardinal${\displaystyle \pi }$ being inaccessible (in some given model of${\displaystyle \mathrm{Z}\mathrm{F}}$ containing${\displaystyle \pi }$) is${\displaystyle {\mathrm{\Pi }}_1}$.^[5]

• Worldly cardinal, a weaker notion
• Mahlo cardinal, a stronger notion
• Club set
• Inner model
• Von Neumann universe
• Constructible universe

• Large cardinals

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