Proving the Irrationals Are Uncountable

Real Analysis · Axiom Academy

EXAMPLE Proving Is Uncountable A proof by contradiction that turns two known facts — is countable, is not — into a statement about the irrationals. Prove that the set of irrational numbers is uncountable . You may use two results established earlier: is countable, and is uncountable (Cantor's diagonal argument). You may also use the lemma that the union of two countable sets is countable. You proved an uncountability result without running a diagonal argument of your own — by borrowing one and letting a union lemma carry it across. Uncountability is a negative claim: "no list works" gives nothing to build, so assume a list exists and break it. The union lemma is the lever: since a union of two countable sets is countable, writing an uncountable set as forces X to be uncountable. Density is not size: meets every interval yet is countable, while its complement — which looks like "the leftovers" — is the uncountable part. Result: is uncountable. In the cardinality sense almost every real number is irrational. The same two moves prove more. The algebraic numbers — all roots of polynomials with integer coefficients — are countable, so minus the algebraic numbers is uncountable: almost every real is transcendental, even though exhibiting a single transcendental number takes real work.

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