Abstract
The diagonal argument is a very famous proof, which has influenced many areas of mathematics. However, this paper shows that the diagonal argument cannot be applied to the sequence of potentially infinite number of potentially infinite binary fractions. First, the original form of Cantor’s diagonal argument is introduced. Second, it is demonstrated that any natural number is finite, by a simple mathematical induction. Third, the concept of potential infinity, created by Aristotle, is presented. Typically, the natural numbers are considered potentially infinite. However, although any natural number is finite, there is also no limit to how large a natural number can be. Fourth, the concept of the potentially infinite decimal is introduced. Fifth, it is easily proven that the diagonal argument cannot be applied to the sequence of all n-bit binary fractions in the interval [0,1). Finally, the diagonal argument is shown to be inapplicable to the sequence of the potentially infinite number of potentially infinite binary fractions, which contains all n-bit binary fractions in the interval [0,1) for any n.
Highlights
The basic usage of numbers is counting objects, followed by measuring length
Even though the discovery was a big shock for rationalism, nobody had thought that real numbers were uncountable before the 19th century
Georg Cantor devised the diagonal argument in the 19th century (Cantor, 1890-91), claiming that real numbers were uncountable
Summary
The basic usage of numbers is counting objects, followed by measuring length. For the latter, this requires determining the unit of length. The ancient Greeks discovered a length that did not have a ratio of itself to the unit length. Georg Cantor devised the diagonal argument in the 19th century (Cantor, 1890-91), claiming that real numbers were uncountable. He constructed the theory of transfinite numbers (Cantor, 1952). Kotani sequence of the potentially infinite number of potentially infinite binary fractions, which contains all n-bit binary fractions in the interval [0,1) for any n
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