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The following is attributed to Cantor:

Consider the set of all polynomials with integer coefficients. For each polynomial P, denoted a0 + a1x + a2x^2 +...+ anx^n, let N(P) be the sum of the absolute value of (i+1)ai, i = 0..n. Note that N(P) is a non-negative integer in all cases.

e.g. P = x^7 - x^3 + 2(x^2) + 1 yields N(P) = 8 + 4 + (2*3) + 1 = 19

Now, if N(P)=j, then the degree of P must be less than or equal to j, and the largest coefficient in P must have absolute value less than j. Thus, for each integer j, there are only a finite number of polynomials P such that N(P) = j. Since each of these finite number of polynomials has only a finite number of roots, there are only a finite number of algebraic numbers associated with each integer j. Since there are only aleph-null integers, it follows that are no more than aleph-null algebraic numbers. Since there are aleph-one real numbers, it follows that there exist aleph-one transcendental numbers.

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