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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.
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.
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.
Historical review with comprehensive references:
See also:
The proof that a Tarski-Grothendieck universe is equivalently a set of $\kappa$-small sets for $\kappa$ an inaccessible cardinal is in
Abstract: We consider four notions of strong inaccessibility that are equivalent in ZFC and show that they are not equivalent in ZF.
Last revised on January 22, 2023 at 12:23:38. See the history of this page for a list of all contributions to it.