nLab inaccessible cardinal

Contents

Context

Universes

Foundations

foundations

The basis of it all

 Set theory

set theory

Foundational axioms

foundational axiom

Removing axioms

Contents

Idea

An inaccessible cardinal is a cardinal number κ\kappa which cannot be “accessed” from smaller cardinals using only the basic operations on cardinals. It follows that the collection of sets smaller than κ\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 CardCard well-ordered and indeed indexed by the collection of ordinal numbers OrdOrd. In particular, we assume the law of excluded middle and thus use 22 (instead of Ω\Omega or PropProp) for the set of truth values in the definition.

An inaccessible cardinal is a regular strong limit cardinal. Here, κ\kappa is regular if every sum of <κ\lt\kappa cardinals, each of which is <κ\lt\kappa, is itself <κ\lt\kappa; κ\kappa is a strong limit if λ<κ\lambda\lt \kappa implies 2 λ<κ2^\lambda\lt\kappa. In other words, the class of sets of cardinality <κ\lt\kappa is closed under the operations of indexed unions and taking power sets.

By this definition, 00 (the cardinality of the empty set) and 0\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 00 by interpreting the requirement that 1<κ1 \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, κ\kappa is a weak limit if λ<κ\lambda\lt\kappa implies λ +<κ\lambda^+\lt\kappa, where λ +\lambda^+ is the smallest cardinal number >λ\gt\lambda. Every strongly inaccessible cardinal is also weakly inaccessible, while the converse is true assuming the generalized continuum hypothesis.

Properties

An (uncountable) cardinal κ\kappa is inaccessible precisely when the κ\kappath level V κV_\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 0\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 κ\kappa-small sets for κ\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.

Last revised on September 30, 2022 at 16:08:07. See the history of this page for a list of all contributions to it.