nLab
regular cardinal

Contents

Idea

A regular cardinal is is a cardinal number that is ‘closed under union’. The category of sets bounded by a regular cardinal has several nice properties, making it a universe that is handy for some purposes but falls short of being a Grothendieck universe. Unlike Grothendieck universes (which are based on inaccessible cardinals rather than regular cardinals), it is easy to prove that (even uncountable) regular cardinals exist.

Definitions

An infinite cardinal κ\kappa is a regular cardinal if it satisfies the following equivalent properties:

  • no set (in a material set theory) of cardinality κ\kappa is the union of fewer than κ\kappa sets of cardinality less than κ\kappa.

  • no set (in a structural set theory) of cardinality κ\kappa is the disjoint union of fewer than κ\kappa sets of cardinality less than κ\kappa.

  • given a function PXP \to X (regarded as a family of sets {P x} xX\{P_x\}_{x\in X}) such that |X|<κ{|X|} \lt \kappa and |P x|<κ{|P_x|} \lt \kappa for all xXx \in X, then |P|<κ{|P|} \lt \kappa.

  • the category Set <κ\Set_{\lt\kappa} of sets of cardinality <κ\lt\kappa has all colimits (or just all coproducts) of size <κ\lt\kappa.

  • the cofinality of κ\kappa is equal to κ\kappa.

A cardinal that is not regular is called singular.

Finite regular cardinals?

Traditionally, one requires regular cardinals to be infinite. This clause may be removed, in which case 00, 11, and 22 are all regular cardinals.

Other modifications of the definition which are equivalent for infinite cardinals may include some of 00, 11, and 22 but not all. For instance, if we regard an indexed disjoint union iλα i\sum_{i\in\lambda} \alpha_i as a binary operation taking as input λ\lambda and a λ\lambda-indexed family α\alpha, then closure under this binary operation as in the above definition also entails closure under the ternary version iλ jμ iα i,j\sum_{i\in\lambda} \sum_{j\in \mu_i} \alpha_{i,j}, and so on. The unary version is simply the identity operation, whereas the nullary version will always output the singleton set 11. (This can be seen by thinking in terms of trees of uniform finite height, or remembering that a dependent sum includes a binary cartesian product as a special case, so a nullary dependent sum should at least include a nullary product.) Thus, from this perspective, 22 is a regular cardinal, but 00 and 11 are not. In applications for which this perspective is the relevant one, such as familial regularity and exactness, one may more precisely be interested in an arity class rather than a regular cardinal.

We may rule out all three finite regular cardinals by additionally generalising from indexed disjoint unions to finitary disjoint unions.

Then in terms of Set <κSet_{\lt\kappa}, the (potential) conditions on a (possibly finite) regular cardinal are as follows:

  1. Set <κSet_{\lt\kappa} is closed under iterated disjoint unions ( iA i\biguplus_i A_i).
  2. Set <κSet_{\lt\kappa} is closed under the nullary iterated disjoint union (the singleton).
  3. Set <κSet_{\lt\kappa} is closed under binary disjoint unions (ABA \uplus B).
  4. Set <κSet_{\lt\kappa} is closed under the nullary disjoint union (the empty set).

These are all variations on the theme of closure under disjoint unions.

Clauses (2–4) hold of all infinite cardinals, while clauses (2&3) together force κ\kappa to be greater than any finite cardinal. However, if we require only clauses (1&2), then 22 is a regular cardinal.

In weak foundations

Thinking of a regular cardinal as a cardinal number makes the most sense using the axiom of choice. Otherwise, we probably want to think of it as a collection of cardinals, or equivalently think of it as the category Set <κSet_{\lt\kappa}.

From this perspective, a regular cardinal is a full subcategory of SetSet that is closed under taking quotient objects and satisfies the condition on Set <κSet_{\lt\kappa} above. We can then recover κ\kappa as the smallest cardinal number greater than every cardinal in Set <κSet_{\lt\kappa}, if we accept the axiom of choice.

Note that if we require only conditions (1&2) on Set <κSet_{\lt\kappa}, then (even classically), {1}\{1\} is an acceptable (and finite) regular collection of cardinals, even though it is not actually of the form Set <κSet_{\lt\kappa} for any cardinal number κ\kappa.

In the absence of the axiom of choice, it is not clear that there exist arbitrarily large regular cardinals. Thus in weaker foundations, regular cardinals (or “regular sets of cardinals”) can be regarded as a large cardinal property.

At least if “regular cardinal” has its classical meaning of a particular ordinal, then the statement that there exist arbitrarily large regular cardinals is independent of ZF; in fact it is consistent with ZF that all uncountable cardinals are singular. A foundational axiom which is related to the existence of regular cardinals (but considers them as sets with various closure properties, rather than cardinal numbers) is the regular extension axiom.

Examples

Regular cardinals

  • The first (infinite) regular cardinal is 0=||\aleph_0 = {|\mathbb{N}|}, because a set with cardinality less than 0\aleph_0 is a finite set, and a finite union of finite sets is still a finite set.

  • The successor of any infinite cardinal, such as 1\aleph_1, is a regular cardinal. (This requires the axiom of choice.) In the case of 1\aleph_1, this means that a countable union of countable sets is countable. Note that this implies that there exist arbitarily large regular cardinals: for any cardinal λ\lambda there is a greater regular cardinal, namely λ +\lambda^+.

Singular cardinals

  • ω= n n\aleph_\omega = \bigcup_{n\in \mathbb{N}} \aleph_n is singular, more or less by definition, since n< ω\aleph_n\lt\aleph_\omega and ||= 0< ω{|\mathbb{N}|} = \aleph_0 \lt\aleph_\omega.

  • More generally, any limit cardinal that can be “written down by hand” should be singular, since if it were regular then it would be weakly inaccessible, and the existence of weakly inaccessible cardinals cannot be proven in ZFC (if ZFCZFC is consistent). We say ‘should’ rather than ‘must’, since there are exceptions, but they are sort of cheating: one (definable with Choice) is ‘the smallest limit cardinal that is regular if and only if some weakly innaccessible cardinal exists’.

  • Assuming the consistency (with ZFC) of ‘there is a proper class of strongly compact cardinals?’, it is consistent with ZFZF that every uncountable cardinal is singular (and in fact every infinite well-orderable cardinal has cofinality 0\aleph_0), a result due to Moti Gitik?. (Of course this conclusion is inconsistent with ZFCZFC, in which many uncountable cardinals, starting with 1\aleph_1, are regular.)

Revised on June 23, 2013 21:15:26 by Tim Porter (95.147.236.182)