Intervals, Uncountability, and Decimals

The rationals and the reals are both infinite and both dense, yet they are not the same size: $Q$ is countable and $R$ is not. Two independent proofs establish that $R$ is uncountable, and decimal expansions, built as suprema of finite truncations, account for both why $0.4999 \dots = 0.5000 \dots$ and why the rationals are precisely the eventually-repeating expansions. The power-set diagonal argument reappears, applied to the digits of real numbers.

Intervals

For $a < b$ in $R$ , the four bounded intervals are

[a, b] = {x : a \leq x \leq b}, (a, b) = {x : a < x < b},

(a, b] = {x : a < x \leq b}, [a, b) = {x : a \leq x < b},

with $[a, b]$ closed, $(a, b)$ open, and the mixed forms half-open. Allowing an endpoint to be $\pm \infty$ gives the unbounded intervals $[a, \infty)$ , $(a, \infty)$ , $(- \infty, b]$ , $(- \infty, b)$ , and $(- \infty, \infty) = R$ . What unifies all nine types is a single order property: an interval is a set with at least two points that contains everything between any two of its members.¹

This no gaps description is what makes intervals the right domains for the intermediate value theorem later on: a connected piece of the line and an interval are the same thing.

All intervals have the same cardinality, regardless of length. The map $x \mapsto 2 x$ is a bijection $[0, 1] \to [0, 2]$ , and $x \mapsto tan x$ is a bijection $(- π /2, π /2) \to R$ . So a bounded open interval has the same cardinality as the entire real line. Cardinality does not detect length, the property that distinguishes $[0, 1]$ from $[0, 2]$ ; measuring length requires a different theory, developed much later.

ℝ is uncountable: nested intervals

The rationals are countable — they can be listed. The reals cannot, and the first proof is Cantor's original argument of 1874. It is cleanest stated as a direct construction: any countable subset of $R$ that is dense must still miss a point.¹

Proof. Let $X = {x_{1}, x_{2}, x_{3}, \dots}$ be any countable subset of $R$ dense enough that every open interval meets it. (If $R$ were countable we could take $X = R$ .) Build two sequences of endpoints. Set $a_{1} = x_{1}$ , $b_{1} = x_{1} + 1$ , so $x_{1} \in / (a_{1}, b_{1})$ . Having chosen $a_{k - 1} < b_{k - 1}$ , let $a_{k} = x_{n}$ for the smallest $n$ with $x_{n} \in (a_{k - 1}, b_{k - 1})$ , and let $b_{k} = \frac{a _{k} + b _{k - 1}}{2}$ . Then

a_{k - 1} < a_{k} < b_{k} < b_{k - 1}, x_{1}, \dots, x_{k} \in / (a_{k}, b_{k}) .

The nested open intervals $(a_{1}, b_{1}) \supset (a_{2}, b_{2}) \supset \dots$ each exclude one more term of the list. Every $a_{n}$ is below every $b_{m}$ , so by the separated-sets proposition $y := sup {a_{n}}$ exists and satisfies $a_{n} < y < b_{n}$ for all $n$ . Thus $y \in (a_{n}, b_{n})$ while $x_{n} \in / (a_{n}, b_{n})$ , forcing $y \neq = x_{n}$ for every $n$ . So $y \in / X$ , and $X$ misses a real. No list exhausts $R$ . $■$

The argument rests on the fact that a decreasing chain of closed intervals in $R$ always has a common point (a direct consequence of completeness), whereas in $Q$ such a chain can close in on a gap and meet nowhere. This nested interval property is a recurring form of completeness, equivalent to the least-upper-bound axiom.

A decreasing chain of closed intervals in the complete line always shares a single common point; the same chain in the rationals could close on a gap.

Decimal expansions from scratch

Decimals are the everyday name for reals, but $0. d_{1} d_{2} d_{3} \dots$ needs a definition — an infinite string of digits is not obviously a number. The definition uses completeness: a decimal is the supremum of its finite truncations. Restricting to $[0, 1]$ loses nothing, since adding an integer part recovers the rest of the line. A digit is an integer in ${0, 1, \dots, 9}$ , and for a digit string $0. d_{1} d_{2} d_{3} \dots$ the $n$ -th truncation is

D_{n} = \frac{d _{1}}{10} + \frac{d _{2}}{1 0 ^{2}} + \dots + \frac{d _{n}}{1 0 ^{n}} .

Two facts make this a genuine correspondence between strings and numbers in $[0, 1]$ .²

Part (1) holds because $D_{n}$ is bounded above by the all-nines string, whose truncations sum by the geometric formula to $1 - 1 0^{- n} < 1$ , so the supremum exists in $[0, 1]$ by completeness. Part (2) constructs the digits greedily: at each step the Archimedean property supplies the largest digit keeping $D_{n} < x$ , and the resulting truncations converge up to $x$ because $x - D_{n} \leq 1 0^{- n}$ can be made arbitrarily small. The construction is a shrinking-interval process: choosing $d_{1}$ selects one of ten subintervals of length $1 0^{- 1}$ , choosing $d_{2}$ selects one of ten sub-subintervals of length $1 0^{- 2}$ , and the nested intervals close on the single point $x$ .

Each decimal digit selects one of ten equal subintervals; the nested choices form a chain of shrinking intervals converging to the represented number.

Non-uniqueness of expansions

The greedy construction in part (2) enforces the strict inequality $D_{n} < x$ , and that is what makes its representation unique. Dropping the strict requirement admits a second string for certain numbers.

The only numbers with two representations are those that terminate — expansions ending in all $0$ s, which also admit an all- $9$ s form. Every other real has exactly one decimal expansion. The ambiguity occurs in any positional system, and it is precisely the case $D_{n} = x$ where a truncation hits the number exactly.

ℝ is uncountable: the decimal diagonal

With representations in hand, Cantor's better-known 1891 proof is a short diagonal argument — the same argument used for the power set, now reading the diagonal digits of a claimed list.²

The decimal diagonal: read the circled diagonal digit of each listed number and change it, producing a number differing from every row.

Rationals are the repeating decimals

Decimals also detect rationality exactly. Long division of $p / q$ produces a remainder in ${0, 1, \dots, q - 1}$ at each step; with only finitely many possible remainders, some remainder recurs, and from that point the digits cycle.²

The converse holds too (an eventually periodic decimal is a rational, summing a geometric series), so the rationals coincide with the eventually-repeating expansions. Every non-repeating decimal, such as $0.101001000100001 \dots$ , is therefore irrational, which gives an easy supply of irrational numbers and a second route to the uncountability of the irrationals: the repeating strings are countable, but the strings are not.

The Cantor set

Digit expansions in base three yield a standard example of an uncountable set of length zero. Fix base three, so a number in $[0, 1]$ is $0. t_{1} t_{2} t_{3} \dots$ with each $t_{j} \in {0, 1, 2}$ , and remove at every stage the points whose expansion needs the digit $1$ . Concretely, delete the open middle third of each remaining interval and repeat.

The first stages of the middle-thirds construction; each step deletes the open middle third of every remaining segment, leaving the Cantor set in the limit.

What remains is the set of numbers whose base-three expansion uses only $0$ and $2$ . Mapping each such expansion to the binary string obtained by halving every digit gives a bijection with all binary strings, i.e. with $P (N)$ , so the Cantor set is uncountable — as large as $R$ itself. Yet its total length is zero.

The Cantor set returns as a standard source of counterexamples in the study of metric spaces and integration.

Lebl, Basic Analysis I, §1.4 — Intervals and the size of $R$ : the nine interval types and their order characterization, the equal cardinality of all intervals, and Cantor's 1874 nested-interval proof that $R$ is uncountable. ↩ ↩²
Lebl, Basic Analysis I, §1.5 — Decimal representation of the reals: decimals as suprema of truncations, existence and uniqueness of representations with the strict bound, the terminating-number ambiguity, the decimal diagonal proof of uncountability, and the eventual periodicity of rational expansions. The middle-thirds set is the base-three analogue of this representation machinery. ↩ ↩² ↩³

Intervals

ℝ is uncountable: nested intervals

Decimal expansions from scratch

Non-uniqueness of expansions

ℝ is uncountable: the decimal diagonal

Rationals are the repeating decimals

The Cantor set

Footnotes