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

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), $1$ (the cardinality of the point), and $\aleph_0$ (the cardinality of the set of natural numbers) are all inaccessible. Usually one explicitly requires inaccessible cardinals to be uncountable, so as to exclude these cases. One can also justify excluding $0$ and $1$ 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 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.

References

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.