A document by Ron Ragusa. Click on the document to view its contents.

In this paper we will see how by varying the initial conditions of Cantor's Diagonal Argument we can use the method to produce a one-to-one correspondence between the set of natural numbers and the set of infinite binary decimals in the open interval (0, 1). We concede that the initial list of infinite binary decimals is, in fact, incomplete and that the diagonal method does produce a number not contained in the list. Also, we'll agree that there are an infinite number of binary decimal numbers in the interval that aren't in the list. We will see how using the same diagonal method we can create infinitely many binary decimal numbers not initially contained in the interval and that each number we so create will correspond with one and only one natural number.

A document by Ron Ragusa. Click on the document to view its contents.

In 1891 Georg Cantor published his Diagonal Method which, he asserted, proved that the real numbers cannot be put into a one-to-one correspondence with the natural numbers. In this paper we will see how by varying the initial conditions of Cantor’s proof we can use the diagonal method to produce a one-to-one correspondence between the set of natural numbers and the set of infinite binary decimals in the interval (0, 1). In the appendix we demonstrate that using the diagonal method recursively will, at the limit of the process, fully account for all the infinite binary decimals in (0, 1). The proof will cement the one-to-one correspondence between the natural numbers and the infinite binary decimals in (0, 1).

In 1891 Georg Cantor published his Diagonal Argument which, he asserted, proved that the real numbers cannot be put into a one-to-one correspondence with the natural numbers.In this paper we will see how by varying the initial conditions of the demonstration we can use Cantor’s method to produce a one-to-one correspondence between the set of natural numbers and the set of infinite binary decimals in the open interval \((0, 1)\).

Interval Sieve Algorithm Creating a Countable Set of Real Numbers from a Closed Interval © 2019 by Ron Ragusa      All rights reserved