nLab inaccessible cardinal

Idea

An inaccessible cardinal is a cardinal number $\kappa$ which cannot be “accessed” from smaller cardinals using only the basic operations on cardinals. It follows that the collection of sets smaller than $\kappa$ satisfies the axioms of set theory.

Definition

The discussion here makes sense in the context of the axiom of choice, since only then is the collection of cardinal numbers $Card$ well-ordered and indeed indexed by the collection of ordinal numbers $Ord$. In particular, we assume the law of excluded middle and thus use $2$ (instead of $\Omega$ or $Prop$) for the set of truth values in the definition.

An inaccessible cardinal is a regular strong limit cardinal. Here, $\kappa$ is regular if every sum of $\lt\kappa$ cardinals, each of which is $\lt\kappa$, is itself $\lt\kappa$; $\kappa$ is a strong limit if $\lambda\lt \kappa$ implies $2^\lambda\lt\kappa$. In other words, the class of sets of cardinality $\lt\kappa$ is closed under the operations of indexed unions and taking power sets.

By this definition, $0$ (the cardinality of the empty set) and $\aleph_0$ (the cardinality of the set of natural numbers) are both inaccessible. Usually one explicitly requires inaccessible cardinals to be uncountable, so as to exclude these cases. One can also justify excluding $0$ by interpreting the requirement that $1 \lt \kappa$ as the nullary part of a requirement whose binary part is closure under indexed unions.

A weakly inaccessible cardinal is a regular weak limit cardinal; sometimes inaccessible cardinals are called strongly inaccessible in contrast. Here, $\kappa$ is a weak limit if $\lambda\lt\kappa$ implies $\lambda^+\lt\kappa$, where $\lambda^+$ is the smallest cardinal number $\gt\lambda$. Every strongly inaccessible cardinal is also weakly inaccessible, while the converse is true assuming the generalized continuum hypothesis.

Properties

An (uncountable) cardinal $\kappa$ is inaccessible precisely when the $\kappa$th level $V_\kappa$ of the von Neumann hierarchy is a Grothendieck universe (Williams), and hence in particular itself a model of axiomatic set theory. For this reason, the existence of inaccessible cardinals cannot be proven in ordinary axiomatic set theory such as ZFC. The axiom asserting that there exists an inaccessible (which amounts to the existence of a Grothendieck universe) is thus the beginning of the study of large cardinals.

If one thinks of $\aleph_0$ as already an inaccessible cardinal, then the axiom of infinity may be seen as itself the first large cardinal axiom.

Generalisations

In Boolean toposes

The concept of an inaccessible cardinal can be generalised to any Boolean topos with choice. Given an Boolean topos $\mathcal{E}$ where all epimorphisms are split, the set of cardinals $\mathrm{Card}(\mathcal{E})$ is the quotient of the groupoid of objects $\mathrm{Ob}(\mathcal{E})$ by mere existence of an isomorphism, with the cardinality function being the canonical quotient map $t \mapsto \vert t \vert:\mathrm{Ob}(\mathcal{E}) \to \mathrm{Card}(\mathcal{E})$.

$$\forall x, y \in \mathrm{Ob}(\mathcal{E}), \vert x \vert = \vert y \vert \iff \exists f \in \mathrm{Hom}_\mathcal{E}(x, y).(\exists g \in \mathrm{Hom}_\mathcal{E}(y, x).g \circ f = \mathrm{id}_x) \wedge (\exists h \in \mathrm{Hom}_\mathcal{E}(y, x).f \circ h = \mathrm{id}_y)$$

The same quotient operation leads to a poset structure on $\mathrm{Card}(\mathcal{E})$ induced by the monomorphisms in $\mathcal{E}$, and $\mathcal{E}$ being a Boolean topos implies that the poset structure on $\mathrm{Card}(\mathcal{E})$ is a decidable total order with its negation being a pseudo-order. As a result, the same definitions of a regular cardinal, a strong limit cardinal, and an inaccessible cardinal in set theory can be made in $\mathrm{Card}(\mathcal{E})$.

An object $x \in \mathrm{Ob}(\mathcal{E})$ is called an inaccessible object if its cardinality is inaccessible.

Related concepts

• regular cardinal

• product-regular cardinal

• Grothendieck universe

• large cardinal

References

Historical review with comprehensive references:

• Ralf Krömer, §6.4.5 in: Tool and object: A history and philosophy of category theory, Science Networks. Historical Studies 32 (2007) [doi:10.1007/978-3-7643-7524-9]

See also:

• Tom Leinster, Large Sets 9, web

The proof that a Tarski-Grothendieck universe is equivalently a set of $\kappa$-small sets for $\kappa$ an inaccessible cardinal is in

• N. H. Williams, On Grothendieck universes, Compositio Mathematica, 21:1 (1969) (numdam)

• Andreas Blass, Ioanna M. Dimitriou, Benedikt Löwe, Inaccessible cardinals without the axiom of choice, Fund. Math. 194 (2007) 179-189 pdf

Abstract: We consider four notions of strong inaccessibility that are equivalent in ZFC and show that they are not equivalent in ZF.