Page "Georg Cantor" Paragraph 26

Cantor established these results using two constructions.

His first construction shows how to write the real algebraic numbers as a sequence a < sub > 1 </ sub >, a < sub > 2 </ sub >, a < sub > 3 </ sub >, ....

In other words, the real algebraic numbers are countable.

Cantor starts his second construction with any sequence of real numbers.

Using this sequence, he constructs nested intervals whose intersection contains a real number not in the sequence.

Since every sequence of real numbers can be used to construct a real not in the sequence, the real numbers cannot be written as a sequence — that is, the real numbers are not countable.

By applying his construction to the sequence of real algebraic numbers, Cantor produces a transcendental number.

Cantor points out that his constructions prove more — namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.

Cantor's next article contains a construction that proves the set of transcendental numbers has the same " power " ( see below ) as the set of real numbers .< ref > Cantor's construction starts with the set of transcendentals T and removes a countable subset