countable set


Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

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

syntax object language

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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idemponent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels


Countable sets


Let SS be any set, and let N\mathbf{N} be the set of natural numbers. Let |S||S| be the cardinality of SS, and let 0\aleph_0 be the cardinality of N\mathbf{N}.

Then SS is:

  • denumerable if |S|= 0{|S|} = \aleph_0,
  • countable if |S| 0{|S|} \leq \aleph_0, and
  • uncountable if |S|> 0{|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:

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

Of course, the terms are even less standardised here.

Examples and Properties

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 (