The real numbers: Attempts to understand

Epple writes in [2]:-
What was a real number at the end of the 19th century? An intuitive, geometrical or physical quantity, or a ratio of such quantities? An aggregate of things identical in thought? A creation of the human mind? An arbitrary sign subjected to certain rules? A purely logical concept? Nobody was able to decide this with certainty. Only one thing was beyond doubt: there was no consensus of any kind.
Were the real numbers consistent? Would an inconsistency appear one day and much of the mathematical building come tumbling down? Some of the intuitive difficulties that began to be felt revolved around the fact that the real numbers were not countable, that is, they could not be put in 1-1 correspondence with the natural numbers. Cantor proved that the real numbers were not countable in 1874. He produced his famous "diagonal argument" in 1890 which gave a second, more striking, proof that the real numbers were not countable. To do this he assumed that the real numbers were countable, that is they could be listed in order. Suppose that this list is
L={n1,n2,n3,n4,...}L = \{n_{1} , n_{2} , n_{3} , n_{4} , ... \}
and let d(i,j)d(i, j) be the ii-th digit of njn_{j} . Define the real number rr to have kk-th digit 1 if d(k,k)=2d(k, k) = 2 and rr to have kk-th digit 2 if d(k,k)2d(k, k) ≠ 2. Then the real number rr is not in the list LL since if it were then it would be ntn_{t} for some tt. But the tt-th digit of rr differs from the tt-th digit of ntn_{t} by construction, so we have a contradiction. Hence the real numbers are not countable.

We'll construct a certain real number which, although historically not one that was looked at, will let us understand some of the questions that arose. Let us start with the 100 two digit numbers. A simple code will let us translate these into letters, 00 become aa, 01 become bb, ... , 25 becomes zz, 26 becomes AA, 27 becomes BB, ... , 51 becomes ZZ, then code all the punctuation marks, and then make all the remaining numbers up to 99 translate to an empty space. Now create a number, say cc, starting from the 100 2-blocks.
cc = 0.01020304050607080910111213141516171819202122232425...
Then continue with the 10000 pairs of 2-blocks 0000, 0001, 0002, ..., 0099, 0100, 0101, ...

Then the 1000000 triples of 2-blocks etc. We can represent cc as a point on a line segment of length 1. Yet every English sentence ever written or ever to be written, occurs in the decoding of cc into letters. For example "one third" has 9 characters so will be decoded from cc around 101810^{18} digits after the decimal point. This article is there, both with the misprints which inevitably occur and a corrected version is there (but one has to go rather a long way to the right of the decimal point to find it!). The whole of Shakespeare is there, as is every book yet to be written, etc!

Let us use cc to describe a paradox which was discovered in 1905. The first thing to notice is that all descriptions of real numbers in English (let us forget about words in other languages) must appear in cc, since every possible sentence occurs in cc. For example, "one third" will occur as we have noted, as will "the base of natural logs", and "the ratio of the circumference of a circle to its diameter", etc. This will enable us to explain Richard's paradox, discovered by Jules Richard in 1905. There are only a countable number of such descriptions of real numbers in English so all real numbers (except a tiny countable subset) can never be described in English. However, this is not the paradox. Richard obtained that by using Cantor's diagonal argument. Create a real number rr, say, as follows:

If the nn-th block of cc translates into a description of a real number r(n)r(n) then set the nn-th digit of rr to be different to the nn-th digit of r(n)r(n). If the nn-th block of cc does not describe a real number (most of course will not even be meaningful in English) then set the nn-th digit of r(n)r(n) to be 1.

Now the real number rr cannot be described in English, since it differs by construction from every real number which can be described in English. That is a bit worrying. Even worse, of course, is the fact that we have just described rr in English in the previous paragraph! If Richard's paradox tells us anything then perhaps it is a warning not to use English (or any other language for that matter) when we are doing mathematics.

Emile Borel introduced the concept of a normal real number in 1909. His idea was to provide a test as to whether the digits in a real number occurred in the sort of way they would if we chose each one at random. First assume that we have a real number written in base 10, that is a decimal expansion. Then if it is a "random" number the digit 1 should occur about 1 /10 of the time so, if we denote by N(1,n)N(1,n) the number of times 1 occurs in the first nn decimal digits, then N(1,n)/nN(1,n)/n should tend to 1 /10 as nn tends to infinity. Similarly for the all digits ii in the set {0, 1, 2, ..., 9) we should have N(i,n)/nN(i,n)/n tending to 1 /10 as nn tends to infinity. But a specific 2-digit number, say 47, should occur among all two digit blocks about 1 /100 of the time etc. Borel called a number normal (in base 10) if every kk-digit number occurred among all the kk digit blocks about 1/10k1/10^{k} of the time. He called a number absolutely normal if it was normal in every base bb.

Borel was able to prove that, in one sense, almost every real number was normal. His proof of this involved showing that the non-normal numbers formed a subset of the reals of measure zero. There were still an uncountable number of non-normal numbers, however, which was easily seen by taking the subset of all real numbers with no digit equal to 1. These are uncountable as can be seen using Cantor's diagonal argument, but clearly they are all non-normal. Clearly no rational is normal since eventually it ends in a repeating pattern. However despite proving these facts, Borel couldn't show that any specific number was absolutely normal. This was achieved first by Sierpinski in 1917.

In his list of problems which he proposed to the International Congress of Mathematicians at Paris in August 1900, Hilbert stated that one of the most pressing issues for the foundations of mathematics was a proof of the consistency of arithmetic. He attempted to solve this himself but was unsuccessful. It was one thing trying to prove arithmetic consistent, but it was known that set theory led to paradoxes. These paradoxes worried many mathematicians and they felt that the foundations of mathematics needed to be built on a logical foundation which did not contain inherent contradictions. Three major paradoxes were due to Burali-Forti in 1897, Russell in 1902, and Richard in 1905. The first of these was derived from the fact that the ordinal numbers themselves formed an ordered set whose order type had to be an ordinal number. Russell's paradox is the well-known one relating to the set of all sets which do not contain themselves as an element, and Richard's paradox we have explained above. Solutions proposed by some mathematicians would only allow mathematics to treat objects which could be constructed. Poincaré (1908) and Weyl (1918) complained that analysis had to be based on a concept of the real numbers which eliminated the non-constructive features. Weyl argued in this 1918 work that analysis should be built on a countable continuum. It was the uncountable, and so non-constructible, aspects of the real line which Weyl felt caused problems.

Gödel proved some striking theorem in 1930. He showed that a formal theory which includes the arithmetic of the natural numbers had to lead to statements which could neither be proved nor disproved within the theory. In particular the consistency of arithmetic was unprovable unless one used a higher order system in which to create the proof, the consistency of this system being equally unprovable. In 1936 Gentzen proved arithmetic consistent, but only by using transfinite methods which were less accepted than arithmetic itself. Although this topic of research is still an active one, most mathematicians accept the uncountable world of Cantor and the non-constructive system of real numbers.

In 1933 David Champernowne, who was an undergraduate at Cambridge University and a friend of Alan Turing, devised Champernowne's number. Write the numbers 1, 2, 3, ..., 9, 10, 11, ... in turn to form the decimal expansion of a number
0.12345678910111213141516171819202122232425262728293031323334353637383940 ...
In The Construction of Decimals Normal in the Scale of Ten published in the Journal of the London Mathematical Society in 1933, Champernowne proved that his number was normal in base 10. In 1937 Mahler proved that Champernowne's number was transcendental. In fact he proved the much stronger result that if p(x)p(x) is a polynomial with integer coefficients then the real number obtained by concatenating the integers p(1),p(2),p(3),..p(1), p(2), p(3), ... to get
is transcendental. Champernowne's number is the special case where p(x)=xp(x) = x. In 1946 Copeland and Erdős proved that the number
0.2357111317192329313741434753596167717379838997101103107109113127131137139 ...
obtained in a similar way to Champernowne's number, but using primes instead of all positive integers, was normal. Neither Champernowne's number nor the Copeland and Erdős number is absolutely normal.

It is reasonable to ask whether π, √2, ee etc are normal. The answer is that despite "knowing" that such numbers must be absolutely normal, no proof of this has yet been found. In fact although no irrational algebraic number has yet been proved to be absolutely normal nevertheless it was conjectured in 2001 that this is the case.

In 1927 Borel came up with his "know-it-all" number. We illustrate this by again using our number cc. Since cc contains every English sentence, it contains every possible true/false question that can be asked in English. Create a real number kk as follows. If the nn-th block of cc translates into a true/false question then set the nn-th digit of kk to be 1 if the answer to the question is true, and 2 if the answer is false. If the nn-th block of cc does not translates into a true/false question then set the nn-th digit of k to be 3. Then kk answers very possible question that has ever been asked, or ever will be asked, in English. Borel describes kk as an unnatural real number, or an "unreal" real. Borel devotes a whole book [1], which he published in 1952, to discuss another idea, namely that of an "inaccessible number".

An accessible number, to Borel, is a number which can be described as a mathematical object. The problem is that we can only use some finite process to describe a real number so only such numbers are accessible. We can describe rationals easily enough, for example either as, say, one-seventh or by specifying the repeating decimal expansion 142857. Hence rationals are accessible. We can specify Liouville's transcendental number easily enough as having a 1 in place n!n! and 0 elsewhere. Provided we have some finite way of specifying the nn-th term in a Cauchy sequence of rationals we have a finite description of the resulting real number. However, as Borel pointed out, there are a countable number of such descriptions. Hence, as Chaitin writes in [6]:-
Pick a real at random, and the probability is zero that it's accessible - the probability is zero that it will ever be accessible to us as an individual mathematical object.
In 1936 Turing published a paper called On computable numbers. Rather than look at the real numbers which can be described in English, Turing looked at a very precise description of a number, namely one which can be output digit by digit by a computer. He then took Richard's paradox and ran through it again, this time with computable numbers. Clearly computer programs, being composed from a finite number of symbols, are countable. Hence computable numbers are countable. List all computer programs -- in fact they will all occur in the number cc above. Create a new real number tt by Cantor's diagonal argument whose nn-th digit is defined as follows. If the nn-th block is a program which outputs a real number, make the nn-th digit of tt different from the nn-th digit of the computable number which is output. If the nn-th block is not a valid programme to output a real number, then make the nn-th digit of tt equal to 1. Now tt cannot be computable since, by construction, it differs from each computable number in at least one digit. However, we have just given a recipe to produce the number which could easily be programmed to output tt, so tt is computable.

Although the "English descriptions" of Richard's paradox must hold the key to the paradox, in this case our "computable numbers" are very precise and not subject to the same difficulties. Do we really have the ultimate paradox which shows that the real numbers are inconsistent? No! So where is the error in our paradox? The error lies in the fact that when we run the computer programmes we do not know whether they will ever output an nn-th digit. We can deduce from this argument that it is impossible to tell whether a computer programme which has output kk digits will ever output a k+1k+1-st digit.

References (show)

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Written by J J O'Connor and E F Robertson
Last Update October 2005