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The Real Numbers are Denumerable

James Edwin Rock

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Abstract

We show Cantor’s diagonal argument has an invalid premise. We create a non-hierarchical Level Set Theory by setting 1/(Aleph Null) = 0. The real numbers have the same cardinality as the set of natural numbers, since the power set of the natural numbers has the same cardinality as the natural numbers. Using binary decimals, we define r, 0 ≤ r ≤ 1 as the ratio of ones to the total number of digits for decimals in the closed interval [0, 1]. We use these ratios to show that the real numbers are denumerable.

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What this paper is about

We show Cantor’s diagonal argument has an invalid premise. We create a non-hierarchical Level Set Theory by setting 1/(Aleph Null) = 0. The real numbers have the same cardinality as the set of natural numbers, since the power set of the natural numbers has the same cardinality as the natural numbers. Using binary decimals, we define r, 0 ≤ r ≤ 1 as the ratio of ones to the total number of digits for decimals in the closed interval [0, 1]. We use these ratios to show that the real numbers are denumerable.

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Available abstract

We show Cantor’s diagonal argument has an invalid premise. We create a non-hierarchical Level Set Theory by setting 1/(Aleph Null) = 0. The real numbers have the same cardinality as the set of natural numbers, since the power set of the natural numbers has the same cardinality as the natural numbers. Using binary decimals, we define r, 0 ≤ r ≤ 1 as the ratio of ones to the total number of digits for decimals in the closed interval [0, 1]. We use these ratios to show that the real numbers are denumerable.

Key concepts: Countable set, Natural number, Cardinality (data modeling), Mathematics, Cantor's diagonal argument, Uncountable set, Aleph, Cardinal number (linguistics)

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