nLab countable unions of countable sets are countable



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

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) 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





(a countable union of countable sets is countable, aka the countable union theorem)

Assuming the axiom of countable choice then:

Let II be a countable set and let {S i} iI\{S_i\}_{i \in I} be an II-dependent set of countable sets S iS_i. Then the disjoint union

iIS i \underset{i \in I}{\cup} S_i

is itself a countable set.


Classical proof: we may assume all the S iS_i are nonempty. For each iIi \in I, choose a surjection f i:S if_i: \mathbb{N} \to S_i (this requires the axiom of countable choice) and also a surjection f:If: \mathbb{N} \to I. Then we have a composite surjection

pair×f×1I×iIiIf iiIS i\mathbb{N} \stackrel{pair}{\to} \mathbb{N} \times \mathbb{N} \stackrel{f \times 1}{\to} I \times \mathbb{N} \cong \underset{i \in I}{\cup} \mathbb{N} \stackrel{\underset{i \in I}{\cup} f_i}{\to} \underset{i \in I}{\cup} S_i

where for pair:×pair: \mathbb{N} \to \mathbb{N} \times \mathbb{N} we may take for example the function that is inverse to (x,y)(x+y+12)+y(x, y) \mapsto \binom{x+y+1}{2} + y.

For constructive mathematicians who accept the axiom of countable choice, the proof is only slightly more elaborate. Here we define a set to be countable if it is a quotient of (is enumerated by) a decidable subset, i.e., a complemented subobject of \mathbb{N}. Thus, supposing we have chosen surjections out of decidable subsets (f i:J iS i) iI(f_i: J_i \to S_i)_{i \in I}, and a surjection JIJ \to I out of a decidable subset JJ, we have a diagram (switching to \sum to denote disjoint sums)

J × pair 1 f iIJ i iI I iIf i iIS i \array{ & & & & J \cdot \mathbb{N} & \hookrightarrow & \mathbb{N} \cdot \mathbb{N} \cong \mathbb{N} \times \mathbb{N} & \stackrel{pair^{-1}}{\to} & \mathbb{N} \\ & & & & \downarrow \mathrlap{f \cdot \mathbb{N}} & & & & \\ \sum_{i \in I} J_i & \hookrightarrow & \sum_{i \in I} \mathbb{N} & \cong & I \cdot \mathbb{N} & & & & \\ \mathllap{\sum_{i \in I} f_i} \downarrow & & & & & & & & \\ \sum_{i \in I} S_i & & & & & & & & }

whose limit might be denoted jJK j\sum_{j \in J} K_j where K jJ f(j)K_j \coloneqq J_{f(j)}. This is certainly a complemented subobject of \mathbb{N}, the complement being formed as ( jJ¬K j)+(¬J)\left(\sum_{j \in J} \neg K_j\right) + (\neg J) \cdot \mathbb{N}, and the limit obviously maps surjectively onto iIS i\sum_{i \in I} S_i, as desired.

The implication countable choice \Rightarrow countable union theorem cannot be reversed, as there are models of ZF where the latter holds, but countable choice fails. Further, the countable union theorem implies countable choice for countable sets, but this implication also cannot be reversed.


Last revised on January 9, 2023 at 21:33:54. See the history of this page for a list of all contributions to it.