Infinity refers to that which has no end. Inherently filled with paradoxes and contradictions, it is a concept found in math, science, and philosophy, and it can refer to time, space, and numbers.

**Mathematical Infinity**

*Infinite Sets*

In mathematics, infinity is not a number itself but a construct to refer to a sequence of numbers with no ending. The common symbol for infinity, <x>, was introduced into mathematical literature by the English mathematician John Wallis in the 17th century. Positive integers are an example of a set of infinite numbers, because there is no last number. For any number that one can consider, there is always a higher one.

Nineteenth century mathematician Georg Cantor stated that a collection is infinite if some of the parts are as big as the whole. This can seem to create paradoxes. For example, if one has an infinite number of objects, he or she can add or remove objects from the group and still have the same quantity. In the above example, positive integers make up an infinite series, but so do a set of just positive *even *integers. The second set comprises only a part of the first set, which would appear to make it only half as large. But in both cases, a larger number always follows, so they are both infinite.

Aristotle described two ways of looking at an infinite series. In actual infinity, one conceives of the series as completed. In potential infinity the series is never completed, but it is considered infinite because a next item in the series is always possible.

*Other Uses of Infinity in Mathematics*

The concept of infinity was introduced into geometry in the 17th century by Gerard Desargues. He developed projective geometry, in which “infinity” is the point where two parallel lines meet, a concept that was already being used in art.

In the algebraic function *y* = 1/*x, y* approaches infinity as *x* approaches zero. The expression *y *could never actually be infinity, since *x* could never be zero, because an expression of division by zero has no meaning in algebra. However, the smaller that *x* gets, the more times it could be divided into 1.

**Paradoxes of Infinity**

The Greek philosopher Zeno of Elea wrote of the dichotomy paradox. In order to travel a certain distance, we would first need to travel half of the total distance. In turn, before we reach that point, we must first have gotten halfway there (1/4 of the total distance), and so on. The distance we would need to travel first becomes infinitely smaller, so that it is impossible even to get started—for every distance to travel, there is always a smaller distance to travel first. These circumstances obviously would not have the same result in a real-life occurrence. However, the paradox demonstrates the idea of a distance being infinitely divisible. In other words, there are an infinite number of fractions between zero and 1 (or any two numbers).

The same idea is demonstrated in Zeno’s paradox of Achilles and the tortoise. In the story, Achilles, a swift runner, races the slow tortoise but allows him a head start of 100 feet. If we imagine that each racer is going at a constant speed (one fast, one slow), then after some period of time, Achilles would have reached 100 feet—the tortoise’s starting point. But by that time, the tortoise would have advanced a further distance. By the time Achilles catches up to the new point, the tortoise would have gone still farther. Zeno states the paradox: Achilles would never catch up to the tortoise, because the more distance he travels, the further ahead the tortoise is. To catch up to the tortoise in the story, Achilles would need to complete an infinite number of actions. Aristotle responded to this paradox by using the ideas of actual and potential infinity. Like the dichotomy paradox, this paradox assumes an actual infinity of points between Achilles and the tortoise.

Neither of these paradoxes would happen in a real-life occurrence, because a person can travel a given distance without having to take account of each point between the start and finish. Therefore, the traveler would be able to reach the destination, and Achilles would catch up to (and surpass) the tortoise in a finite time.

Bertrand Russell described the Tristram Shandy paradox, which is based on the title character of the novel by Laurence Sterne. In the novel, Shandy attempts to write his autobiography but finds that it takes him 2 years to write about the first 2 days of his life. Shandy complained that he would never be able to complete the writing, because the more he wrote, the more material there would be to write about. Russell suggested that if Shandy lived for an infinite number of years, then every day of his life would eventually be written about.

The paradox lies in the fact that there would never come a day when Shandy could finish the book, because with each day of writing, he became a year farther away from his goal.

**Infinity of Time**

*Time and the Universe*

In some schools of thought, time is inextricably linked to the universe itself. Thus, many theories dealing with the infinity of time are actually referring to the infinity of the universe. Saint Augustine wrote that time exists only within the created universe, with God existing outside of time in the “eternal present.” While science theorizes that the universe began with the big bang, we cannot know if time began at this point as well or if there was time before there was a universe. According to some theories, the universe will end with a big crunch, but again, it is unknown if time would end at that point as well. Richard Tolman, in his oscillatory universe theory, hypothesized that the universe is in an infinite cycle of big bangs followed by big crunches.

*An Infinite Past*

The possibilities of an infinite past or infinite future have been discussed by philosophers for millennia. In 1692, English theologian and scholar Richard Bentley rejected the idea of an infinite past while accepting the idea of an infinite future. He wrote that the revolution of the earth, occurring in the present, could continue into infinity, because the future was inexhaustible. However, an infinite number of revolutions in the past was not possible, because each revolution would have once been part of the present and therefore finite. Immanuel Kant also rejected an infinite past, saying that an infinity cannot be completed, and if the past were infinite, we would never have arrived at the present moment.

**Modern Perspectives**

Debate about various aspects of infinity has continued into the 20th and 21st centuries. Albert Einstein wrote that our universe was finite but had no boundaries. Physicist Stephen Hawking built upon this idea, proposing that time and space together were finite but boundless (similar to the surface of the earth) and therefore had no beginning.

Galileo stated that infinity, by its very nature, was incomprehensible to human minds. Nevertheless, it is likely that infinity, with its possibilities and paradoxes, will continue to be a source of theory and discussion.

*Jaclyn McKewan*

** See also** Aquinas, Saint Thomas; Augustine of Hippo, Saint; Bruno, Giordano; Eternity; Gödel, Kurt; Kant, Immanuel; Russell, Bertrand; Zeno of Elea

**Further Readings**

Clegg, B. (2003). *A brief history of infinity.* London: Robinson.

Craig, W. L. (1978). Whitrow and Popper on the impossibility of an infinite past. *The British Journal for the Philosophy of Science, 30,* 165-170.

Lavine, S. (1998). *Understanding the infinite. *Cambridge, MA: Harvard University Press.

Whitrow, G. J. (1980). *The natural philosophy of time.*

New York: Oxford University Press.