In set theory, a cardinal number is a strongly inaccessible cardinal if it is uncountable, regular, and a strong limit cardinal. A cardinal is a weakly inaccessible cardinal if it is uncountable, regular, and a weak 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 by . Strongly inaccessible cardinals were introduced by and as Grenzzahlen (English "limit numbers").

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

. 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 of Zermelo-Fraenkel set theory, even with the axiom of choice (ZFC), for which no inaccessible cardinals exist. On the other hand, it is also believed that there are models of ZFC for which even strongly inaccessible cardinals do exist. That ZFC can accommodate these large sets, but does not necessitate them, provides an introduction to the large cardinal axioms. See also Models and consistency.

The existence of a strongly inaccessible cardinal is equivalent to the existence of a Grothendieck universe is a strongly inaccessible cardinal then the von Neumann stage. As expected from their correspondence with strongly inaccessible cardinals, Grothendieck universes are very well-closed under set-theoretic operations.

An ordinal are regular ordinals, but not limits of regular ordinals.)

From some perspectives must be a weakly inaccessible cardinal.

Models and consistency

is a cardinal number. Zermelo–Fraenkel set theoryth level of the Von Neumann universe is a model is strongly inaccessible. Furthermore, ZF implies that the Gödel universe 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.

, then

1. is one of the intended models of Zermelo–Fraenkel set theory;

2. 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;

3. is one of the intended models of Morse–Kelley set theory.

(see constructible universe to be a standard model of ZF (see below).

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 .

Existence of a proper class of inaccessibles

There are many important axioms. 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.

, i.e. the class of all ordinals in your model.

α-inaccessible cardinals and hyper-inaccessible cardinals

-inaccessible-weakly inaccessible. In this case the 0-weakly inaccessible cardinals are the regular cardinals and the 1-weakly inaccessible cardinals are the weakly inaccessible cardinals.

ththth 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-inaccessible.) It is occasionally used to mean Mahlo cardinal.

-hyper-inaccessible-inaccessible-hyper-inaccessible.

Hyper-hyper-inaccessible cardinals and so on can be defined in similar ways, and as usual this term is ambiguous.

-hyper-inaccessible".

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

Two model-theoretic characterisations of inaccessibility

has the following reflection is an elementary substructure is closed unbounded-indescribableth inaccessible cardinal.

) cannot, due to Tarski's theorem.

Secondly, under ZFC Zermelo's categoricity theorem is a model of second order ZFC.

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.

,.

Without the axiom of choice

In ZF without the axiom of choice, we can define what it means to be strongly inaccessible. An ordinal κ is strongly inaccessible iff ω < κ and for all α < κ, for every function from Vα to κ, there is β < κ such that the image of the function is a subset of β.

Similarly, an ordinal κ is weakly inaccessible iff ω < κ and for all α < κ, for every function from α+ to κ, there is β < κ such that the image of the function is a subset of β. Here α+ is the successor cardinal (i.e. Hartogs number) of α.

See also

• Worldly cardinal, a weaker notion

• Mahlo cardinal, a stronger notion

• Club set

• Inner model

• Von Neumann universe

• Constructible universe

Works cited

• . English translation: .

References