nLab
countable unions of countable sets are countable

Contents

Context

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

semantics

Contents

Statement

Proposition

(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.

Proof

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.

References

Last revised on April 26, 2019 at 12:41:04. See the history of this page for a list of all contributions to it.