Infinity (mathematics)

Summary: Infinity is an important part of the curriculum and has a rich and interesting history.

Counting comes naturally to humans. Children as young as 2 years old begin to associate numbers with groups of objects: 1, 2, 3, 4, and 5 are quickly understood. The concept of “plus one” also develops early in life. Given any whole number, there is always a “next number,” the one achieved by adding one. As such, early in life we face the reality that there is no largest number, for given a number of any size, adding one to it produces a number that is yet bigger. That is, the set of natural numbers {1, 2, 3, 4, 5,…} is infinite. While the concept of infinity is fundamental in mathematics, cosmology, and theology, many of the advances in understanding infinity were met with severe criticism or worse. For example, according to some stories, Hippasus, a member of the Pythagorean order, was drowned for divulging the existence of infinite non-repeating decimals. Revolutions in philosophy and mathematics resolved many of the fascinating paradoxes related to infinity, but infinity continues to challenge and interest us today.

A Hotel Example

Infinity is a concept, but it is not itself a number. To illustrate how the notion of infinity is different, it is helpful to turn to one of the great mathematicians of all time, David Hilbert (1862–1943). In 1900, he spoke to the International Congress of Mathematicians about 23 unsolved problems that he considered to be the most important to the progress of mathematics—the search for solutions to these problems shaped a great deal of twentieth-century mathematics, and some even remain open to this day. Besides being a leading mathematician, Hilbert was also a thoughtful teacher, and he was reputed to have used the following paraphrased story to challenge his students to think about the curious nature of infinity.

A mathematician owned an unusual hotel, one with infinitely many rooms. Each room was assigned a natural number—Room 1, Room 2, and so on—and on one occasion, it happened that every room in the hotel was filled. A customer seeking a room walked into the lobby and asked the manager if there were any openings. The manager reported that every room was full but that there was a way for the customer to get a room.

The occupant of Room 1 was asked to move to Room 2; the occupant of Room 2 moved to Room 3; and in general, the person in Room n stepped next door to Room N+1. The customer who had requested a room at the entirely full hotel was now able to occupy Room 1.

The next day, when the hotel was still completely full, an unusual charter bus arrived, carrying infinitely many passengers, all seeking rooms. At first, the members of this group were disheartened to learn that the hotel was completely booked. But the mathematically savvy manager once again had a solution.

The occupant of Room 1 was asked to move to Room 2; the occupant of Room 2 moved to Room 4; the person in Room 3 went to Room 6; and in general, the person in Room n stepped down the hall to Room 2N. The customers getting off the bus were now able to move into all of the odd-numbered rooms, as rooms 1, 3, 5, 7…were all open.

While this story may seem far-fetched because there are only a finite number people alive on Earth, it illustrates some remarkable properties of natural numbers and raises concerns, such as whether more natural numbers exist than there are natural numbers.

Infinite Sets

Georg Cantor’s revolutionary ideas on the sizes of such infinite sets form the basis of many ideas in modern mathematics, including the fields of analysis and calculus. For example, removing the odd natural numbers from the set of all natural numbers

{1, 2, 3, 4, 5, 6… 2n, 2n + 1,…}

leaves the set {2, 4, 6… 2n, 2n + 2,…}

which is yet another infinite set. Galileo Galilei believed that the sizes of infinite sets could not be compared or contrasted. However, Cantor and mathematicians today agree that since a first even natural number can be identified, a second even natural number, and so on, just as a first natural number can be identified, a second natural number, and a third, then there are the same number of even natural numbers as there are natural numbers since they can be put in one-to-one correspondence. Cantor also proved that there are uncountable sets that have a different measure of infinity, such as the real numbers. However, Cantor did not receive the recognition during his lifetime that he has today. Some theologians believed his work challenged the uniqueness and infinity of God, and both mathematicians and theologians strongly objected to his work at the time.

Limits

A question that has intrigued many people over the centuries is whether or not the numbers 1 and 0.⁹̅ are the same. In fact they are, as the following argument shows. If we consider the number 0.⁹̅, observe the following:

Certainly, two numbers can be added, three numbers, four numbers, and indeed, as many finite numbers as likened be. From this, observe the following:

Since this last sequence of numbers converges to the number 1, one concludes that the infinite sum is 1. That is,

At first glance, this may seem strange to a person unaccustomed to the role of limits in mathematics. But, as was perhaps first understood by Archimedes in antiquity, limits are the bridge from the finite to the infinite, and they are indispensable to mathematics and the mathematician. Understanding infinity allows for the understanding that the numbers 1 and 0.⁹̅ are the same.

Paradoxes

Certainly the infinity concept presents some challenges and unusual situations. Greek philosopher Zeno of Elea was known for posing paradoxes that challenged mathematicians for centuries. For instance, in Zeno’s Paradox, a person walks toward a wall by each time stepping half the remaining distance, thus taking time stepping half the remaining distance, thus taking an infinite number of steps but (theoretically) never actually reaching the wall.

Another example is Gabriel’s Horn, an infinite surface that can be easily generated by revolving a simple curve about an axis. Interestingly, the surface is not named after its discoverer, Italian physicist and mathematician Evangelista Torricelli, but is rather named after the Archangel Gabriel in order to connect the infinite with theology. This infinite surface can be shown to contain finite volume yet have infinite surface area. In other words, Gabriel’s Horn, if filled with paint, would require only a finite volume, yet that paint could not cover the surface of the horn. While situations like these initially seem impossible, mathematics provides interesting and satisfying explanations of these phenomena.

Modern Developments

In the twentieth and twenty-first centuries, mathematicians continue to grapple with the concept of infinity. French mathematicians Émile Borel, René Baire, and Henri Lebesgue explored rationalist ideas, while a group of Russian mathematicians led by Dmitry Egorov linked mathematics to philosophy and theology. Building upon the French work and using mystical insights gained during their religious practice of Name Worshipping, they founded descriptive set theory, which transformed mathematical analysis.

However, this did not resolve the contradictions of infinitesimals in calculus, which Sir Isaac Newton, Gottfried Leibniz, and Bishop Berkeley had wrestled with during the development of that subject in the seventeenth and eighteenth centuries. Abraham Robinson created the field of nonstandard analysis in 1960 when he gave a rigorous definition of an infinitesimal number, and mathematicians continue to explore the implications of both standard and non-standard analysis. Besides there being infinitely many natural numbers, there are even infinitely many prime numbers. Primes form the building blocks of numbers and in many ways the very foundation of mathematics. In a similar way, calculus rests upon the notion of limit, which at its core involves infinite processes. Because so much of the subject naturally involves the infinite, mathematicians have had to face, understand, and conquer infinity; more than this, the presence of infinity in the world guarantees that there will always be more mathematics to explore, discover, and comprehend.

Bibliography

Clegg, Brian. A Brief History of Infinity: The Quest to Think the Unthinkable. London: Robinson, 2003.

Graham, Lauren. Naming Infinity: A True Story of Religious Mysticism and Mathematical Creativity. Cambridge, MA: Belknap Press of Harvard University Press, 2009.

Rucker, Rudy. Infinity and the Mind. Princeton, NJ: Princeton University Press, 2004.

Stillwell, John. Roads to Infinity: The Mathematics of Truth and Proof. Natick, MA: A K Peters, 2010.