Paradoxes in Mathematics by Stanley J. Farlow

Author:Stanley J. Farlow [Stanley J. Farlow]
Language: eng
Format: epub, mobi
Publisher: Dover Publications, Inc.
Published: 2014-06-26T16:00:00+00:00

Finger matching between 1,2,3,... and the positive fractions

Figure 3

If the table in Figure 3 were continued indefinitely downward and to the right, it would contain all the rational numbers 1, ½, ⅛, … (with many duplications, such as 2/2 and 3/3, 1/3 and 3/9, and so on), Hence, Cantor concluded natural numbers and rational numbers each had the same number of elements.2 In other words, each had cardinality . Like we said, things get strange in Cantor’s world of infinity.

Larger Infinities? Cantor then looked around for sets that might have “more” members, the most obvious set to try being the real numbers. The answer was astounding. Cantor proved that there are “more” real numbers than natural numbers: 1,2,3,…, that it is impossible to match up the real numbers with the natural numbers, and that since the natural numbers are a subset of the real numbers, the cardinality of the real numbers must be larger than that of the natural numbers. So how did Cantor do this?

Proofs that show something cannot happen are often proved by what is called proof by contradiction, where one assumes the contrary, that it can happen. In this case, we assume the rational numbers and the real numbers between 0 and 1 can be matched up in a one-to-one manner, and then arrive at some type of contradiction, from which our conclusion is there is no correspondence between the rational numbers and these real numbers.

Cantor used an ingenious proof called the diagonalization process, where he hypothesized that it was possible to match up the natural numbers with all positive real numbers. A typical matching is shown in Figure 4 where the natural numbers 1,2,3,… are in the left column, and typical real numbers, expressed in decimal form, on the right.

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