Statement¶

Let $C \subset [0,1]$ be the middle-thirds Cantor set, the attractor of the iterated function system $f_0(x)=x/3$, $f_1(x)=x/3+2/3$. Then:

The similarity (Hausdorff) dimension of $C$ is $$D = \frac{\log 2}{\log 3} \approx 0.6309297535714574.$$
$C$ has Lebesgue measure zero: $\lambda(C)=0$.
$C$ is uncountable, with cardinality $2^{\aleph_0}=\mathfrak{c}$.

Assumptions¶

$C$ is the unique nonempty compact attractor of the two contractions $f_0,f_1$, each with ratio $r=1/3$ (so $N=2$).
Open set condition (OSC): the open set $U=(0,1)$ satisfies $f_0(U)\cup f_1(U)\subseteq U$ with $f_0(U)\cap f_1(U)=\varnothing$ (the images $(0,1/3)$ and $(2/3,1)$ are disjoint). This is what licenses equating the similarity dimension with the Hausdorff dimension.
Lebesgue measure $\lambda$ on $\mathbb{R}$ is used for part 2; base-3 (ternary) expansions for part 3.

Proof¶

[asserted] By Moran's theorem / Hutchinson's theorem, an IFS of similarities satisfying the OSC has an attractor whose Hausdorff dimension $D$ is the unique solution of the Moran equation $\sum_{i=1}^{N} r_i^{D}=1$. For $N=2$ equal ratios $r=1/3$ this is $2\cdot(1/3)^{D}=1$.
[verified] Solving $2\cdot 3^{-D}=1$ gives $3^{D}=2$, hence $D=\log 2/\log 3$. SymPy solve(2*(1/3)**D - 1, D) returns log(2)/log(3), numerically $0.6309297535714574$.
[verified] Measure zero. At step $k$ one removes $2^{k}$ open intervals each of length $3^{-(k+1)}$. The total removed length is the geometric series $$\sum_{k=0}^{\infty}\frac{2^{k}}{3^{k+1}} =\frac{1}{3}\sum_{k=0}^{\infty}\Big(\tfrac{2}{3}\Big)^{k} =\frac{1}{3}\cdot\frac{1}{1-2/3}=1.$$ SymPy summation(2**k/3**(k+1),(k,0,oo)) returns 1. Since the removed set has full measure $1$ in $[0,1]$, $\lambda(C)=1-1=0$.
[asserted] Uncountability. A point of $[0,1]$ lies in $C$ iff it has a base-3 expansion using only the digits $\{0,2\}$ (the removed middle thirds are exactly the points forced to have digit $1$; endpoints with a terminating expansion also admit a $\{0,2\}$ form). The map $(d_1,d_2,\dots)\mapsto \sum_k d_k 3^{-k}$ is a bijection between $\{0,2\}^{\mathbb{N}}$ and $C$. Since $\{0,2\}^{\mathbb{N}}$ has cardinality $2^{\aleph_0}=\mathfrak{c}$, $C$ is uncountable. This is Cantor's cardinality theorem applied to the ternary address space; it is a theorem-cited conclusion, not a numerical computation.
[verified] Parts 2 and 3 together show $C$ is an uncountable set of measure zero — the defining paradox of the Cantor "dust".

Verification¶

Dimension (step 2): SymPy solved $2\cdot 3^{-D}=1$ symbolically to log(2)/log(3); float value $0.6309297535714574$. Confirmed against the Moran-equation root solver for $N=2,\ r=1/3$.
Measure zero (step 3): SymPy summation(2**k/3**(k+1),(k,0,oo)) = 1 (closed-form geometric series), so the complement in $[0,1]$ has measure zero.
Cardinality (step 4): the $\{0,2\}$-digit characterisation of $C$ is a standard finite-prefix combinatorial fact; the continuum cardinality of $\{0,2\}^{\mathbb{N}}$ follows from Cantor's theorem — asserted from the cited 1883 result, not numerically re-derived.
Asserted external result: the Moran/Hutchinson OSC ⇒ Hausdorff-dimension identity (step 1). All quantitative outputs machine-checked.

Historical source¶

Georg Cantor introduced the set in the fifth installment of Über unendliche, lineare Punktmannigfaltigkeiten (Mathematische Annalen 21, 1883) as an example of a perfect, nowhere-dense set. The dimension identification via the similarity/Moran equation is later (Hausdorff 1919; Moran 1946).

Citations¶

G. Cantor, "Über unendliche, lineare Punktmannigfaltigkeiten V", Mathematische Annalen 21 (1883), 545–591.
P. A. P. Moran, "Additive functions of intervals and Hausdorff measure", Proc. Cambridge Philos. Soc. 42 (1946), 15–23.
J. E. Hutchinson, "Fractals and self-similarity", Indiana Univ. Math. J. 30 (1981), 713–747.
K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 3^rd ed., Wiley (2014), §2.2, §9.1.