Cantor's first set theory article

Contrary to what most mathematicians believe, Georg Cantor's first proof that the set of all real numbers is uncountable was not his famous diagonal argument, and did not mention decimal expansions or any other numeral system.

Suppose a set R is

• linearly ordered, and
• densely ordered, i.e., between any two members there is another, and
• has no "endpoints", i.e., smallest or largest members, and
• has no gaps, i.e., if it is partitioned into two sets A and B in such a way that every member of A is less than every member of B, then there is a boundary point c, so that every point less than c is in A and every point greater than c is in B.

Then R is uncountable.

That is the theorem Cantor proved in December 1873, and published in 1874.

The proof begins by assuming some sequence x₁, x₂, x₃, ... has all of R as its range. Define two other sequences as follows:

a₁ = x₁.

b₁ = x_i, where i is the smallest index such that x_{i is not equal to a1.}

a_n+1=x_i, where i is the smallest index greater than the one considered in the previous step such that x_i is between a_n and b_n.

b_n+1=x_i, where i is the smallest index greater than the one considered in the previous step such that x_i is between a_n+1 and b_n.

The two monotone sequences a and b move toward each other. By the "gaplessness of R, some point c must lie between them. The claim is that c cannot be in the range of the sequence x, and that is the contradiction. If c were in the range, then we would have c = xi for some index i. But then, when that index was reached in the process of defining a and b, then c would have been added as the next member of one or the other of those two sequences, contrary to the assumption that it lies between their ranges.