Two opinions on the infinity

A physicist

Is there a greatest number, i.e. a number greater than any other number? The answer is in the negative. Indeed, the positional system of notation, the system we are accustomed to "since ever", provides a particularly easy proof. By writing a zero to the right of a string representing an integer we multiply it by 10, and such zeros can be added ad infinitum.

Humanity had to cope with problems related to infinity since its very beginnings. Quite soon infinity represented by an infinite process (for instance, the infinity referred to in the previous paragraph, or in the childish and I have one more!) was seen to be different from the infinity corresponding to a quantity of some object (like the number of points on a straight line). The former, by the way, was admitted with less pain.

It was believed at first that infinity cannot be counted. Later, this restriction was made less strict. In fact, we can count natural numbers since we always know which number should succeed any one of them. Nevertheless, it would be erroneous to suppose that infinity can be observed in nature, although such a belief is still held by many. The examples quoted to support this belief usually include the grains of sand on a desert, the drops of water in an ocean or the stars in the sky. The reason is simple. In fact, no human can ever count the number of water drops in an ocean nor the number of sand grains in a desert. There would not be enough time for that!

Well, but is infinity indeed necessary in practice? What has nature to do with very large natural numbers? Using exponential notation we can easily write down a very large number indeed, like e.g. 10^{100}. In the decimal system this is a unit with one hundred zeros. Let's see how big this number is in comparison with the idea of infinity drawn from nature in the examples mentioned above. Let a drop of water have a diameter of 2 mm. This means that its volume is about 4 cubic millimeters. The volume of water contained in all the oceans is estimated at 1370 million square kilometers, so the number of drops "only" amounts to about 3 . 10^{26} . For a mathematician used to the infinity this number does not seem to be very large, even if it is written in the form of a unit followed by twenty six zeros. we can try to express the volume of oceanic waters by the number of water molecules. One drop represents approximately 2 . 10^{-4} moles of water, so it contains about 2 . 10^{19} molecules. This yields 6 . 10^{45} molecules of water in all the oceans. Still not very much when compared to the number 10^{100}. The area of the Earth is "just" 5 . 10^{20} square millimeters. Even the volume of the whole universe does not exceed 10^{100} cubic millimeters (and what about cubic angstroms?). Such computations, however, have no physical significance. There is no such amount of anything in this world and the number 10^{100} is by far greater than anything that can be counted or measured. Here we can observe the difference between physics and mathematics. For a mathematician this huge number is quite the same as, say, 5 or 28, in the sense that it is just one element of the infinite set of finite numbers. The physical reality or nonreality of notions is of no concern to a contemporary mathematician. He lives in a world of abstractions, where the correctness of a deduction and the consistency of a theory is more important than their relation to reality.

The notion of infinity has acquired a very precise meaning in mathematics. The first infinity - an infinitely long process - has its own symbol, something like an eight laid down, introduced into mathematics by an English mathematician, John Wallis, in 1655. It is usually called potential infinity.

On the other hand, there are (infinitely) many infinities of the second kind and they also have their own, very rich and developed, system of notation.

However, from a practical point of view, the number 10^{100} is the infinity. Whoever thinks of it in this way has, in fact, good intuition.

Jan KALINOWSKI
Department of Physics, Warsaw University
kalino@fuw.edu.pl

A mathematician

Just above you can read the article of Jan Kalinowski. No one would probably argue with his opinion that from a physical, or rather: commonsense point of view no number greater than 10^{100} is really necessary. Indeed, when we count the number of sand grains in a hypothetical sphere of Archimedes (with a radius equal to the distance between the sun and the visible stars), or the number of water molecules in all the oceans, or the number of years needed to turn a stone made of substance million times harder than diamond and of the volume of 1 cubic kilometer into dust just by rubbing it with the hand once every million years, or even the number of atoms in the whole universe, we always get numbers smaller than 10^{100} - and considerably so.

However, human imagination can be deceitful. Even without such complex means of description as very large time periods, ultrahard rocks or all the atoms of the universe, we can encounter numbers much greater than $10^{100}$ in quite ordinary situations. First of all, let's turn to a simple combinatorial question. How many possible ways to seat 80 passengers in a second-class railway car are there? (As everyone who leaves his permanent domicile during holidays knows, such a car has ten compartments with eight seats each.) School knowledge is sufficient to provide the answer: there are 80! (factorial of 80). This is approximately

much more than the "physical" barrier of 10^{100}.

Next, let's consider the following simple problem. In how many ways can we choose 1000 inhabitants from a total town population of one million and then assign to them one thousand numbered seats in an express train going, say, from Warsaw to Cracow. Again the answer would come as no surprise to a secondary school pupil. The number of possible ways is

i.e. approximately

Six thousand digits are needed to write this number down in the decimal notation.

Think now of a casino in Las Vegas. Assume that once a year a gambler turns up with $ 100 in his pocket and the intention of staking one dollar on red or black until either his gain amounts to $ 1,000,000 or he loses everything.

A third-year student of mathematics who has been exposed to the problem of a gambler's ruin and who can compute the mathematical expectation of a random variable with geometric distribution should have no serious problem in establishing that in the average about 10^{23000} years are needed to see such a gambler leaving the casino with a million dollars in his pockets. (For comparison: the age of the universe is estimated at about 18 . 10^9 years.) Incidentally, to find the number of years needed to make a gambler rich one only has to know how to compute the complete probability of an event and how to add finite and infinite geometric sequences, so why don't you verify the above result by yourself, dear reader?

And then, once we step into the purely mathematical world of constants occurring in various inequalities or, say, in number theory, real monster numbers can be found. I would like to tell you about one of them.

Little more than 150 years ago Catalan formulated a hypothesis that among the powers of natural numbers (with exponents 2, 3, 4, ...) the only two consecutive natural numbers are 8 (the cube of 2) and 9 (the square of 3). In other words, Catalan claimed that the equation

has a unique solution x=m=3, y=n=2.

Catalan's conjecture is, up to now, one of the famous open problems of number theory. A huge step towards the solution has been made in the seventies by a Dutch mathematician, Robert Tijdeman, who proved that if the numbers x,y,m,n solve Catalan's equation (1), then the greatest of them does not exceed some effectively computable constant C. Soon after various estimations of the constant appeared. M. Langevin found in 1976 that

If we were to write the number of digits of the number of digits of the right-hand side of the inequality, we would need much more than 10^{100} digits. The situation would slightly improve, if we assumed that m and n are at least 3. R. Baker proved that the theorem of Thue, Siegel and Roth implies that the quadruple of numbers m, n, x, y which satisfies the equation (1) must also satisfy the following condition: