Transcendental numbers, the continuum and incompleteness

Almost all real numbers are transcendental

Let's put the two chapters together. The set of algebraic numbers is countable; the set $\mathbb{R}$ is not. If the set of transcendental numbers were countable, $\mathbb{R}$ would be the union of two countable sets, hence countable.

$$\mathbb{R} = \left\{\text{algebraic}\right\} \cup \left\{\text{transcendental}\right\}$$

The transcendental numbers therefore form an uncountable set: they are infinitely more numerous than the algebraic ones. This is the famous 1874 argument, which proves the existence of transcendental numbers without exhibiting a single one.

The contrast with the work of Liouville, Hermite and Lindemann is striking: it took pages of analysis to establish the transcendence of $e$ and $\pi$, whereas a count shows in three lines that almost every real number is transcendental.

Set	Cardinality	Within $\mathbb{R}$
$\mathbb{Q}$	countable	dense, but negligible
algebraic	countable	negligible
transcendental	uncountable	most of the line

The continuum hypothesis

Cantor denotes by $\aleph_0$ the cardinality of $\mathbb{N}$ and by $2^{\aleph_0}$ that of $\mathbb{R}$. Is there a set whose cardinality lies strictly between the two? Cantor conjectured that there is not: this is the continuum hypothesis, the first of Hilbert's twenty-three problems in 1900.

The answer is astonishing. Gödel showed in 1940 that the hypothesis cannot be disproved from the usual axioms of set theory; Cohen showed in 1963 that it cannot be proved from them either. The question is undecidable: one can build consistent mathematics with it, and other consistent mathematics without it.

The diagonal as a method

Cantor's technique has spread far beyond set theory. It always consists of building an object that differs from the $n$-th candidate in its $n$-th feature.

Result	What the diagonal constructs
Uncountability of $\mathbb{R}$	a real number absent from every list
Cantor's theorem on $\mathcal{P}(E)$	a subset with no preimage
Undecidability of the halting problem, Turing 1936	a program that no decider classifies
Incompleteness theorems, Gödel 1931	a true but unprovable statement
Russell's paradox	the set of sets that do not belong to themselves

It is one of the rare arguments whose reach cuts across logic, theoretical computer science and the foundations of mathematics in this way.

Classic pitfall

Concluding from the uncountability of $\mathbb{R}$ that "irrational numbers are more numerous, so you are more likely to pick one at random." The intuition is correct, but it belongs to measure theory, not to cardinality: it is the fact that $\mathbb{Q}$ has measure zero that gives the statement a probabilistic meaning.

Second pitfall: talking about "the largest infinity." Cantor's theorem on $\mathcal{P}(E)$ rules out precisely this idea — every cardinal is followed by a strictly larger one.