I have read that Georg Cantor used the slash diagonal argument to show that the set of the real numbers is great than that of natural numbers. The argument is applied for real numbers between 0 and 1 and is as follow:

1. write some real numbers between 0 and 1, one under the other dropping the 0. that appears in the beginning. In this way, one gets a matrix of digits.
2. take the digits in the diagonal, and add 1 to each digit. In this way, you will find another real number that can not be in the list of real numbers that you have in the matrix, because it will differ from any of those by at least one digit: that of the diagonal.
3. in this way, you show that any list of real numbers between 0 and 1 is incomplete, so you can not find an one-to-one correspondence between this set and the set of integer, and this should show that the set of real numbers between 0 and 1 is greater than that of natural numbers.

My question is: if I take the digits in any row of the matrix as being an integer written from right to left, and then I would have an one-to-one correspondence between the set of natural numbers and that of real numbers between 0 and 1 (1 should not be included, since would result in the same integer as 0. This would mean that the set of real numbers between 0 and 1 is as big as the set of natural numbers. Where is the error in my argument?

Copyright © 1996-2010 Alexander Bogomolny

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