Alumni Newsletter: Spring '99

Reflections on the meaning of infinity

Mathematicians tend to believe in infinity since they find it a very useful idea. Euclid proved that there are infinitely many prime numbers. A prime number is a whole number bigger than I which is divisible only by itself and by 1. The first prime numbers are 2,3,5,7,11,13,...etc. As numbers get larger the prime numbers get scarcer and scarcer, essentially because there are more and more numbers which can divide them. But the list of primes never stops. Euclid's proof of this is one of the simplest and most elegant mathematical proofs. We assume that we can make a finite list all of the prime numbers 2,3,5,..., p. Multiply them all together and add I to get N = 2x3x5x ... xp +1. This new number N is either a prime number or divisible by a prime number. But the number N cannot be divisible by any of the numbers on our list since division by any of them always leaves the remainder 1. We are forced to conclude that N is either a prime number itself or that it is divisible by a prime number that is not on our list. In any case, our list must be incomplete, a contradiction.

Galileo discovered one of the most important properties of infinity. When you have an infinity of things, you can throw some of them away and still have exactly as many things as you started with. To see this, take all of the whole numbers and throw away all of the odd numbers. You are left with the even numbers. But every number can be paired with its double. The number I is paired with 2, the number 2 is paired with 4, the number 3 is paired with 6, and so on. Every whole number can be "married" to its double. Hence, the number of whole numbers is exactly the same as the number of even numbers.

The whole numbers constitute the smallest infinity. Infinities of this size are called countable infinities. Examples of countable infinities include all whole numbers, all even numbers, all odd numbers, and all rational numbers. The founder of set theory, Georg Cantor, discovered that the set of all real numbers is a larger infinity than the set of all whole numbers. It is an uncountable infinity.

Calculus students encounter an infinity which is different from what we have been describing above. Adding up an infinite list of positive real numbers (this is called an infinite series) results in two possibilities called convergence and divergence. The infinite series 1/1 + 1/2 + 1/4 +1/8 + 1/16 + 1/32 +... converges to the finite number 2. Adding up another infinite series I + 2 + 3 + 4 + 5 + 6 + 7 +.... results in infinity. The series diverges.

A divergent sum that most calculus students remember is the so-called harmonic series, the sum 1/1 + 1/2 + 1/3 + 1/4 + 115 +.... This sum grows very slowly, a rough approximation to adding up the first 1,000,000,000 of these numbers is 20. In order to get this sum to be over 100, you would have to add up a very, very large number of terms, approximately, I followed by 45 zeros. Nonetheless, by adding up a ridiculously large number of terms, you can make the sum as large as you like, thus showing that the series diverges.

The fact that the harmonic series diverges is related to the fact that there are infinitely many prime numbers. The slow rate of growth of the sum of the harmonic series is related to the scarcity of prime numbers. Further investigation of this leads to what is regarded as the hardest unsolved problem in mathematics, the so-called Riemann Hypothesis. Watch how the divergence of the harmonic series leads to a proof that there must be infinitely many primes. Pick any prime number and call it p. The infinite series I + I/p + I/pl + I/pl +... is called a geometric series and calculus students know that it converges to a finite number, p/(p-1). The distributive law and the fact that every whole number can be factored uniquely into prime numbers leads to the fact that the harmonic series is the product of the list of all of these convergent geometric series, one for every prime number. Since the harmonic series diverges, there must be infinitely many factors. Finitely many convergent factors cannot produce an infinite answer. In other words, there must be infinitely many prime numbers.

For many years, it was an unresolved question as to whether there was an infinity intermediate in size between the infinity of all whole numbers and the infinity of all real numbers. The assertion that there was no infinity between the two was called the Continuum Hypothesis. The question was finally settled in an unusual way. Kurt Godel discovered that the Continuum Hypothesis was consistent with the rest of mathematics. The Continuum Hypothesis cannot be disproved. Paul Cohen showed that the Continuum Hypothesis is independent of the rest of mathematics. The Continuum Hypothesis cannot be proved.

In this brief introduction to infinity, we have tried to explain the concept and to illustrate some of its mathematical properties. Infinities occur very often in formulas describing physical situations. When infinities appear, they model reality not because they actually occur but because the system is breaking down or at least changing in such a fundamental way that the formula can no longer give a complete picture of the situation. Even in the real world, the concept of infinity is useful