# Paradoxes in Mathematics

## Stanley J. Farlow

# Paradoxes in Mathematics

## Stanley J. Farlow

## About This Book

There's more than one way to define a paradox, and this intriguing book offers examples of every kind. Stanley J. Farlow, a prominent educator and author, presents a captivating mix of mathematical paradoxes: the kind with surprising, nonintuitive outcomes; the variety that rely on mathematical sleight-of-hand to impress the unwary observer; and the baffling type with a solution that passes all understanding.

Students and puzzle enthusiasts will find plenty of thought-provoking enjoyment mixed with a bit of painless mathematical instruction among these twenty-eight conundrums. Some of them involve counting, some deal with infinity, and others draw on principles of geometry and arithmetic. None requires an extensive background in higher mathematics. Challenges include The Curve That Shook the World, a variation on the famous Monty Hall Problem, Space Travel in a Wineglass, Through Cantor's Looking Glass, and other fun-to-ponder paradoxes.

## Information

*circa*360

*ā*280 B.C.), one of the first mathematicians to think about the subject, felt there were two varieties of infinity, the

*potential*and the

*actual*. He said the natural numbers 1, 2, 3, ... are

*potentially*infinite since they go on forever and have no greatest number. He felt they were not

*actually*infinite in size since they were not a completed entity.

*subset*or subcollection of all natural numbers 1,2,3,ā¦ that would seem like there are fewer of them, as illustrated in Figure 1.

*line up*the perfect squares as Figure 2 shows, you can match any perfect square with a natural number, and any natural number with a perfect square, suggesting the two sets have the same number of members.

**The Godfather of Infinity: Georg Cantor:**The subject of infinity as we know it today had its origins in the work of the German mathematician. Georg Cantor (1845ā1918), whose groundbreaking insights rest upon a very simple idea (as do many seminal insights). Suppose you are unable to count and would like to know if your hands have the same number of fingers.

*one-to-one correspondence*: every finger on each hand has exactly one kindred soul on the other.

*You may not know how many fingers you have on each hand*,

*but you know they are the same*.

*count*infinite sets, we can tell if they have the

*same*number of members by applying the āfinger ruleā used when determining that our hands had the same number of fingers.

*size*of a set the

*cardinality*of the set, and if a set can be placed in a one-to-one correspondence with the natural numbers 1,2,3,ā¦ the āsizeā of the set is called

*countably infinite*, and these types of sets are said to have

*cardinality*

^{1}

*all*the rational numbers 1, Ā½, ā , ā¦ (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

**Larger Infinities?**Cantor then looked around for sets that might have āmoreā members, the most obvious set to try being the real numbers. The answer was astounding. Cantor proved that there are āmoreā real numbers than natural numbers: 1,2,3,ā¦, 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?

*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.

*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,ā¦ are in the left column, and typical real numbers, expressed in decimal form, on the right.

## Table of contents

*Paradoxes in Mathematics*([edition unavailable]). Dover Publications. Retrieved from https://www.perlego.com/book/113011/paradoxes-in-mathematics-pdf (Original work published 2014)

*Paradoxes in Mathematics*. [Edition unavailable]. Dover Publications. https://www.perlego.com/book/113011/paradoxes-in-mathematics-pdf.

*Paradoxes in Mathematics*. [edition unavailable]. Dover Publications. Available at: https://www.perlego.com/book/113011/paradoxes-in-mathematics-pdf (Accessed: 14 October 2022).

*Paradoxes in Mathematics*. [edition unavailable]. Dover Publications, 2014. Web. 14 Oct. 2022.