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 $<\kappa$ cardinals, each of which is $<\kappa$, is itself $<\kappa$; $\kappa$ is a strong limit if $\lambda <\kappa$ implies $2^{\lambda }<\kappa$. In other words, the class of sets of cardinality $<\kappa$ 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 $1<\kappa$ 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 inacessible cardinals are called strongly inaccessible in contrast. Here, $\kappa$ is a weak limit if $\lambda <\kappa$ implies ${\lambda }^{+}<\kappa$, where ${\lambda }^{+}$ is the smallest cardinal number $>\lambda$. Every strongly inaccessible cardinal is also weakly inaccessible, while the converse is true assuming the continuum hypothesis. A weakly inaccessible cardinal may be strengthened to produce a (generally larger) strongly inaccessible cardinal.

Mike: What does that last sentence mean? It seems obviously false to me in the absence of CH.

Toby: It means that if a weakly inaccessible cardinal exists, then a strongly inaccessible cardinal exists, but I couldn't find the formula for it. Something like ${\beth }_{\kappa }$ is strongly inaccessible if $\kappa$ is weakly inaccessible (note that ${\aleph }_{\kappa }=\kappa$ then), but I couldn't verify that (or check how it holds up in the absence of choice).

Mike: I don’t believe that. Suppose that the smallest weakly inaccessible is not strongly inaccessible, and let $\kappa$ be the smallest strongly inaccessible. Then $V_{\kappa }$ is a model of set theory in which there are weakly inaccessibles but not strong ones. I’m almost certain there is no reason for the smallest weakly inaccessible to be strongly inaccessible.

JCMcKeown: Surely ${\beth }_{\kappa }$ has cofinality at most $\kappa$, so it can’t be regular. Maybe the strengthening involves some forcing or other change of universe? E.g., you can forcibly shift $2^{\lambda }={\lambda }^{+}$ for $\lambda <\kappa$, and then by weak inaccessibility, etc… I think. Don’t trust me. —- (some days later) More than that: since the ordinals are well ordered, if there is any strongly inaccessible cardinal greater than $\kappa$, then there is a least one, say $\theta$. Then $V_{\theta }$ is a universe with a weakly inaccessible cardinal and no greater strongly inaccessible cardinal. Ih! Mike said that already… So whatever construction will have to work the other way around: if there is a weakly inaccessible cardinal that isn’t strongly inaccessible, and if furthermore a weakly inaccessible cardinal implies a strongly inaccessible cardinal, then the strongly inaccessible cardinal implied must be less than $\kappa$. And that sounds really weird.

Properties

A 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 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, tome 21 no 1 (1969) (numdam)