30 Sep 2019
I came across a website written by a Charles Fisher Cooper who writes articles on various subjects, including mathematics. In the section “George Without the Dots ” on his page Cantor’s Diagonal Argument he claims he has a rigorous diagonal proof that overcomes objections to the standard presentation of the proof.
His argument proceeds by assuming that there is a set of ordered pairs, where the first element of each ordered pair is a natural number, and the second is a decimal expansion of a real number where the digits of such real numbers are “randomly selected ”. The author asserts that each such “ordered pair has a counting number for the first number and a real number for the second.” He also asserts that no two such ordered pairs have an identical real number.
He then claims that what he has done is that he has “defined an effective method for generating unique real numbers, each of which can be assigned to a unique counting number.”
He then proceeds to claim that he can define another real number whose digits are different from every real number in the set of ordered pairs that he has described. He states that this new number is given by taking the ith digit of the ith such number and creating a new number by making its ith digit different to the ith digit of the ith such number.
He concludes that there “will always be an extra number not in the original list. So you will always have more real numbers than counting numbers.”
The fallacies in the argument
One obvious fallacy is when he claims that he has “defined an effective method for generating unique real numbers, each of which can be assigned to a unique counting number.”
The fact is that he has not done so. According to his definition, the digits are random, therefore there is no information as to what the first generated real number is, and so you can’t assign the natural number 1 to it. In fact you cannot know what any such real number is, and you cannot know even one of the digits of any such number.
It’s hard to know whether to laugh or despair when one sees arguments like this. Cooper has managed to assume that it is possible to define a one-to-one correspondence of natural numbers to all real numbers within a given mathematical language - which is precisely what Cantor’s diagonal argument proves is impossible. (Footnote: In fact there are fully formal proofs of this, see Fully Formal proofs of the Diagonal proof.)
The irony here is that Cooper’s Platonist assumption that random digits ‘exist’ independently of any unambiguous definition actually demonstrates the fallacy of such Platonist beliefs.
As Cooper himself almost manages to point out, there cannot be any real number that is not included in his description, since all possible sequences of digits are included in his description. So if there could ‘exist’ such a Platonist set of such ordered pairs, there would be an enumeration of all real numbers, and so there would be a contradiction - since there must also ‘exist’ a Platonist diagonal number which is not any of the numbers in the set of ordered pairs, yet every possible sequence of digits already ‘exists’ somewhere in the set of all the ordered pairs.
The notion that there somehow ‘exist’ numbers that have “randomly selected ” digits is a Platonist notion that is lacking in any logical foundation and which leads to contradictions - the random digits are selected by whom? … or what? … some deity perhaps? It’s philosophical nonsense.
See also the post Random Numbers.
While Cooper remarks that his web-page is essentially that which can be read in the book Metalogic: An Introduction to the Metatheory of Standard First Order Logic by Geoffrey Hunter, University of California Press (1973), there does not appear to be any part of the book that corresponds to Cooper’s argument that is discussed above.
It has been pointed out to me that Cooper has since made some additions to his page, including the claim “a quick perusal of Geoffrey’s book (just the first chapter for crying out loud) will show you that the proof given in such painful length and detail above is essentially that which Geoffrey gives us with much less wind.” In fact, no, it isn’t, Geoffrey Hunter’s proof makes no mention whatsoever of ordered pairs or ‘randomly selected’ digits, it is a fairly standard presentation of the diagonal argument. Perhaps Geoffrey Hunter was perspicacious enough to know that adding unnecessary assumptions to his proof would not improve it.
Cooper’s additional blunder
Cooper even manages to make an elementary error in his attempt to show that no two of his ‘randomly selected’ numbers can be identical. He uses conventional probability theory to show that, if two of his numbers are ‘randomly selected’, the theoretical probability that they are identical is zero - which is a correct application of the theory - but then he makes the elementary error of assuming that this is the same theoretical probability that, given one ‘randomly selected’ number, there exists at least one of the infinite quantity of remaining numbers that is identical to that number. That is a completely different matter, and a proof of the former does not constitute a proof of the latter.
In case anyone is having difficulty with understanding this, a simple way of demonstrating this error is by considering the set that contains two copies of every number (this account is simplified for clarity, the case where there are ordered pairs as in Cooper’s account is given in the appendix below). Assume one number is ‘randomly selected’ and removed from the set. Then another number is ‘randomly selected’. As for Cooper’s case, the theoretical probability of the second selected number being identical to the previously selected number is zero, and hence the theoretical probability of these two numbers being identical is zero. (Footnote: Note that the conventional theory of probabilities for the notion of a random selection from an infinite set gives some rather strange results, see Random selection of numbers.)
On the other hand, given that, as before, one number is ‘randomly selected’ and removed from the set, then what is the probability that there remains one number in the set which is identical to the removed number? The answer of course is obvious - it’s a certainty.
This shows that the theoretical probability of two ‘randomly selected’ numbers being identical is quite different to the theoretical probability that there exist two identical numbers.
By the way, Cooper’s concern regarding identical numbers demonstrates that he is completely oblivious to the fact that if there are identical numbers in some enumeration of real numbers, that would not invalidate the diagonal proof anyway, since the diagonal number must still be different to every number in the enumeration.
Appendix - sets of ordered pairs
The case of a set of ordered pairs as in Cooper’s account is readily established. Consider the set that is the union of two sets of ordered pairs. One set has odd natural numbers for the first part of the pair and all natural numbers for the second part of the pair. They other set has even natural numbers for the first part of the pair and all natural numbers for the second part of the pair. Hence in the combined set every natural number occurs twice, as the second part of an ordered pair whose first part is an odd number, and as the second part of an ordered pair whose first part is an even number. Now assume one pair is ‘randomly selected’ and removed from the set. Now another pair is ‘randomly selected’. As for Cooper’s case, the theoretical probability of the number that is the second part of this pair being identical to the second part of the previously selected number is zero, and hence the theoretical probability of these two numbers being identical is zero.
On the other hand, given that, as before, one pair is ‘randomly selected’ and removed from the set, then what is the probability that there remains one pair in the set whose second part is identical to the second part of that removed pair? The answer of course is obvious - it’s a certainty. This shows that the theoretical probability of two selected numbers being identical is quite different to the theoretical probability that there exist two identical numbers.
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