of all homotopy types
An inaccessible cardinal is a cardinal number which cannot be “accessed” from smaller cardinals using only the basic operations on cardinals. It follows that the collection of sets smaller than satisfies the axioms of set theory.
An inaccessible cardinal is a regular strong limit cardinal. Here, is regular if every sum of cardinals, each of which is , is itself ; is a strong limit if implies . In other words, the class of sets of cardinality is closed under the operations of indexed unions and taking power sets.
By this definition, (the cardinality of the empty set), (the cardinality of the point), and (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 and by interpreting the requirement that 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, is a weak limit if implies , where is the smallest cardinal number . Every strongly inaccessible cardinal is also weakly inaccessible, while the converse is true assuming the continuum hypothesis.
A cardinal is inaccessible precisely when the th level 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 as already an inaccessible cardinal, then the axiom of infinity may be seen as itself the first large cardinal axiom.
The proof that a Tarski-Grothendieck universe is equivalently a set of -small sets for an inaccessible cardinal is in
We consider four notions of strong inaccessibility that are equivalent in 𝖹𝖥𝖢 and show that they are not equivalent in 𝖹𝖥.