The Proofs
To prove that the set of real algebraic numbers is countable, Cantor starts by defining the height of a polynomial of degree n to be: n − 1 + |a0| + |a1| + … + |an|, where a0, a1, …, an are the coefficients of the polynomial. Then Cantor orders the polynomials by their height, and orders the real roots of polynomials of the same height by numeric order. Since there are only a finite number of roots of polynomials of a given height, Cantor's orderings put the real algebraic numbers into a sequence.
Next Cantor proves his second theorem: Given any sequence of real numbers x1, x2, x3, … and any interval, one can determine a number in that is not contained in the given sequence.
To find such a number, Cantor builds two sequences of real numbers as follows: Find the first two numbers of the given sequence x1, x2, x3, … that belong to the interior of the interval . Designate the smaller of these two numbers by a1, and the larger by b1. Similarly, find the first two numbers of the given sequence belonging to the interior of the interval . Designate the smaller by a2 and the larger by b2. Continuing this procedure generates a sequence of intervals, … such that each interval in the sequence contains all succeeding intervals. This implies the sequence a1, a2, a3, … is increasing, the sequence b1, b2, b3, … is decreasing, and every member of the first sequence is smaller than every member of the second sequence.
Cantor now breaks the proof into two cases: Either the number of intervals generated is finite or infinite. If finite, let be the last interval. Since at most one xn can belong to the interior of, any number belonging to the interior besides xn is not contained in the given sequence.
If the number of intervals is infinite, let a∞ = limn → ∞ an. At this point, Cantor could finish his proof by noting that a∞ is not contained in the given sequence since for every n, a∞ belongs to the interior of but xn does not.
Instead Cantor analyzes the situation further. He lets b∞ = limn → ∞ bn, and then breaks the proof into two cases: a∞ = b∞ and a∞ < b∞. In the first case, as mentioned above, a∞ is not contained in the given sequence. In the second case, any real number in is not contained in the given sequence. Cantor observes that the sequence of real algebraic numbers falls into the first case, thus indicating how his proof handles this particular sequence.
Read more about this topic: Cantor's First Uncountability Proof
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