countable set

**natural deduction** metalanguage, practical foundations

**type theory** (dependent, intensional, observational type theory, homotopy type theory)

**computational trinitarianism** = **propositions as types** +**programs as proofs** +**relation type theory/category theory**

Let $S$ be any set, and let $\mathbf{N}$ be the set of natural numbers. Let $|S|$ be the cardinality of $S$, and let $\aleph_0$ be the cardinality of $\mathbf{N}$.

Then $S$ is:

**denumerable**if ${|S|} = \aleph_0$,**countable**if ${|S|} \leq \aleph_0$, and**uncountable**if ${|S|} \gt \aleph_0$.

Note that the first two terms are not entirely standardised; some authors use them interchangeably for one or the other concept.

If you accept the axiom of choice, then there is really nothing more to say than what was above. In weaker foundations, more care may be needed. The following seem to be the usual definitions in constructive mathematics:

- $S$ is
**denumerable**if there exists a bijection from $\mathbf{N}$ to $S$. - $S$ is
**countable**if there exists a surjection from a decidable subset of $\mathbf{N}$ to $S$. - $S$ is
**uncountable**if, given any function $f$ from a decidable subset of $\mathbf{N}$ to $S$, the function is strongly non-surjective in the sense that there exists an element of $S$ that is not in the image of $f$.

Of course, the terms are even less standardised here.

The set of real numbers is uncountable. (Arguably, set theory as such begins with Georg Cantor's proof of this statement.)

Given any infinite set, its power set is uncountable by Cantor's theorem.

The empty set is countable.

Any uncountable set must be inhabited.

Any (Kuratowski)-finite set is countable.

Any uncountable set must be infinite.

A denumerable set is precisely an infinite countable set; sometimes this is written as a *countably infinite set*.

countable unions of countable sets are countable

In classical mathematics a countable set is either finite or denumerable. This need not hold in constructive mathematics. We do have, however, that a countable set is either empty or inhabited, which is classically trivial but need not hold constructively for every set.

In some forms of constructive mathematics, especially in the Russian school, it is assumed (or provable from other assumptions) that *every* set is a subset of a countable set. The fact the the set of real numbers is uncountable still applies, however, as the inclusion map of the subset need not split. In particular, the set of computable numbers is a subset of a countable set, but to prove that it is itself countable requires excluded middle.

Revised on May 18, 2017 14:12:53
by Urs Schreiber
(217.92.118.241)