Copyright © James R Meyer 2012 - 2018 www.jamesrmeyer.com
In the book, What is Mathematics? by Courant and Robbins, (Footnote: Courant & Robbins, What is mathematics? pub: Oxford University Press, New York, 2nd ed. 1996, Ch 4, section 2, pp 82-83. ISBN10: 0195105192. (First edition, 1941) What is Mathematics?: details.) there is a proof that there are more real numbers than natural numbers, a proof that, intriguingly, is not based on one of Cantor’s well-known proofs. It appears that it was Courant and Robbins who originally came up with the proof, although it can be seen mentioned elsewhere. (Footnote: Sondheimer & Rogerson, Numbers and Infinity, pub: Cambridge University Press, 1st ed, 1981, ISBN10: 0521284333, Ch 11, pp 152-153 Numbers and Infinity: details.
Gregory Chaitin has made several references to the proof, see:
a) Chaitin, Algorithmic Information Theory, pub: Cambridge University Press, 2004, ISBN13: 9780521616041 (Section 7.1), this part of the book is based on Chaitin’s 1987 paper ‘Incompleteness theorems for random reals’.
b) Chaitin, Thinking about Gödel and Turing, Essays on Complexity, 1970-2007, pub: World Scientific Publishing Co. Pte. Ltd. ISBN13 9789812708953 (Section 2.2: Alternate proof: Any countable/denumerable set of reals has measure zero.)
c) Chaitin, Meta Math, The Quest for Omega, pub: Pantheon, First Edition, 2005, ISBN13: 9780375423130 (Chapter Five, Section: ‘Reals are uncomputable with probability one’)
John Stillwell, in the book Roads to Infinity: The Mathematics of Truth and Proof (CRC Press, 2010 Roads to Infinity: details) claims that Carl Gustav Axel von Harnack was the first to come up with this proof in 1885; however he provides no reference to where one might find Harnack’s proof.)
It is easy to demonstrate that the argument is inherently flawed. However, the interesting thing about it is that a logical analysis of the underlying argument shows that the Platonist belief in the independent ‘actual’ existence of limitlessly many mathematical entitles is inherently illogical, because it shows that that belief leads to a contradiction. The demonstration of the contradiction in the proof by Courant and Robbins serves to demonstrate the untenability of the Platonist viewpoint.
The Courant and Robbins proof goes like this:
It’s almost embarrassingly easy to show that this argument results in a contradiction, which shows that it cannot possibly be correct. It is well known that the rational numbers (numbers that are fractions) between 0 and 1 can be listed (See Appendix: Listing the rational numbers and One-to-one correspondences). They can be set in a list so that for any natural number, there is a corresponding fraction between 0 and 1, and every rational number between 0 and 1 is included in this list. Since that is the case, we can apply the same argument as in the proof above to the fractions between 0 and 1 (note that some people have suggested that they can circumvent the contradiction by using a list of the rationals that are defined in terms of various conditional requirements - this argument is easily dealt with, see Appendix: A specific listing of rational numbers below).
And by the argument of the proof, the conclusion is, similarly, that since the list includes every fraction between 0 and 1, then we know that every fraction must be enclosed by one of these line segments 1⁄10, 1⁄100, 1⁄1000 … etc, regardless of how small that segment may be. And, again, according to the argument of the proof, the sum of these line segments cannot be more than 1⁄9, whereas the entire line segment between 0 and 1 is a length of 1. That means that there must be some remaining part of the line that isn’t made up of any combination of any of the line segments 1⁄10, 1⁄100, 1⁄1000 … .
But there are are only two ways there could be points remaining - as line segments and/or as isolated points. There can’t be any line segments remaining - if we suppose that there could be one such line segment, then there has to be a real number at each end of the line segment, and between any two real numbers, there must always be at least one fraction in that line segment - in fact, there will be limitlessly many fractions in that line segment. That leaves the possibility of isolated points - but then that would require that a collection of isolated points, each of which has a length of zero, somehow has a collective length that is greater than zero. This strange notion is essentially what Lebesgue measure theory asks you to accept, see the page Lebesgue measure theory.
But even the abstruse machinations of Lebesgue measure theory cannot rescue Courant and Robbins. The next section explains why.
First, a couple of definitions:
An open interval is an interval that does not include the endpoints that define that interval (for example the open interval whose endpoints are 1⁄3 and 1⁄2 is the set of all points between 1⁄3 and 1⁄2 but not including the points 1⁄3 and 1⁄2).
A closed interval is an interval whose endpoints are included in the interval.
Now, we define our set of ever decreasing intervals like this:
We start with the closed interval between 0 and 1. As above, we take a listing of the rational numbers between 0 and 1, such as that given in Appendix: A specific listing of rational numbers below. (Footnote: See One-to-one correspondences and Listing the rationals.) Then, going through this list of rational numbers, for the first rational we define an open interval 1⁄10 wide with that rational at the midpoint of the interval; our set now includes all the numbers in that interval (but not including the endpoints). For the next number, we define an open interval 1⁄100 wide with that rational at the midpoint of the interval; we add those numbers to our set. For the next number, we define an open interval 1⁄1000 wide with that rational at the midpoint of the interval; we add those numbers to our set. And so on, with each subsequent open interval being 1⁄10 of the length of the previous interval.
So, now, what numbers between 0 and 1 might not be included in some such interval?
We started with a closed interval whose endpoints are rational numbers. At every iteration, we define an open interval whose endpoints are rational numbers - so those rational endpoints are not removed at that iteration. This means that any interval that is left after an iteration must be a closed interval with rational endpoints.
It follows that there cannot be any number not included by some defined interval. For if there were any number remaining, it would have to be within a closed interval with rational endpoints. That is impossible, since those rational endpoints, by the original definition, must be themselves midpoints of some interval that is defined, by the definition, to be in the set.
Note that this more detailed definition only clarifies the situation - if anything, we are removing less points at each iteration, since we are not removing the endpoints. Furthermore, the lengths of the removed intervals are precisely the same as before, since the endpoints have zero length. All that the more detailed definition does is to hammer in the fact that there cannot be any points between 0 and 1 that are not included in some defined interval. (Footnote: For more on this, see the page on Lebesgue measure, Lebesgue measure theory.)
But some people still refuse to accept the logic behind this and try to devise various arguments against it. You can see an example of such an argument by a well-known professor at Fallacy by hidden definition.
So, we now have a contradiction, which I will call the ‘Courant & Robbins contradiction’. (Footnote: Here we do not euphemistically call it a paradox, and by so doing, imply that there just appears to be a contradiction – there is a genuine contradiction here which cannot be brushed aside.)
We have proved that the defined intervals must cover the entire interval 0 to 1, but Courant & Robbins claim that there are still points not covered by any interval, and moreover that the length of that collection of points is 8⁄9. And that this is always the case regardless of the choice of size of the line segments, as long as the initial fraction is less than 1⁄2 (for example, we could choose 1⁄3 or 1⁄57 instead of 1⁄10). Furthermore, they even claim that by choosing a smaller and smaller initial fraction, the total length of points remaining becomes closer and closer to 1!
When we have a contradiction, that means that the argument is flawed, and that at least one step in the argument must be rejected as incorrect. So what is going on? Where is the flawed step in the argument?
The Platonist view is that all real numbers are some sort of ‘actual’ things that ‘exist’ in some sense in some ‘reality’, and that they exist simultaneously in this ‘reality’ - and that between any twp points there ‘exists’ a section of a line that consists of a limitlessness quantity of points, and that all those points simultaneously ‘exist’ in some sense as ‘actual’ things. The Platonist also believes that every line segment of real numbers ‘exists’ in this ‘reality’, and all of these line segments ‘exist’ simultaneously. So, accordingly, from the Platonist viewpoint, in the Courant & Robbins contradiction, all the line segments 1⁄10, 1⁄100, 1⁄1000 … ‘exist’, and all ‘exist’ simultaneously. And the sum of all of these line segments also ‘exists’ and is 1⁄9.
But if you are not a Platonist there is no contradiction, and the notion of a ‘sum’ of a limitless quantity of line segments simply doesn’t arise. To see the reason why there is no contradiction for the non-Platonist, consider the Courant & Robbins argument:
1⁄10 = 1⁄10
1⁄10 + 1⁄100 = 11⁄100
1⁄10 + 1⁄100 + 1⁄1000 = 111⁄1000
1⁄10 + 1⁄100 + 1⁄1000 + 1⁄10000 = 1111⁄10000
1⁄10 + 1⁄100 + 1⁄1000 + 1⁄10000 + 1⁄100000 = 11111⁄100000
We see the pattern quite easily, which is
1⁄10, 11⁄100, 111⁄1000, 11111⁄10000, 11111⁄100000, 111111⁄1000000 …
And similarly, when we try to divide 1 by 9 on a calculator, we get 0.1111111111111111. On your calculator, depending on the display panel or how it operates, you might get a different quantity of 1’s. And if you did the calculation by hand, you can add more and more 1’s the longer you calculate for. Theoretically, there is no limit to the quantity of 1’s you could calculate; but one thing is for sure – you will never calculate more than a finite quantity of 1’s.
If we look at the pattern that is
1⁄10, 11⁄100, 111⁄1000, 11111⁄10000, 11111⁄100000, 111111⁄1000000 …
it leads us to suspect that these values are all approaching a value of 1⁄9. This can be confirmed by a formula given by commonly accepted mathematical logic (See Appendix: Geometric Series) but we would be careful to note 1⁄9 is a limiting value that the values never actually reach. The formula is not the value of the sum of a limitless sum of fractions.
It might be worth pointing out here that, generally speaking, mathematicians aren’t very rigorous when they talk about things that involve limitlessness. Often we see ‘numbers’ written as a sequence of decimal digits with three dots at the end, like this, 0.21382349… The three dots at the end are subject to various interpretations. (Footnote: When it is said, for example, that 0.1326472… is a number, where the … signifies continuing into infinity, in fact this not signify any particular number. What it does actually signify is a set of real number values that lie between the values 0.1326472 and 0.1326473.) Often mathematicians will talk about the ‘sum’ of an infinite series of fractions, when what they actually are referring to is not to an actual sum of an infinite quantity of fractions, but to a limit – a limit that cannot be exceeded by the sum of any finite series of fractions. In most cases, the distinction doesn’t matter, but sometimes it does.
When it comes to the notion (as in the Courant & Robbins proof) that we form a one-to-one correspondence of every line segment to a fraction, what this notion actually involves is a definition which simply defines limitlessly many single correspondences, and there is no limit to the quantity of such correspondences. This definition does not require the ‘existence’ of limitless quantities of fractions or of line segments; it is simply a valid logical definition, and it has no dependence on any Platonist beliefs. So when the Courant & Robbins contradiction states that the sum of a limitless quantity of fractions of the form 1⁄10 + 1⁄100 + 1⁄1000 + 1⁄10000 + 1⁄100000 + … is 1⁄9, the response of a non-Platonist is that that is incorrect. He has no need of any notion such as the sum of a limitless quantity of fractions. For the non-Platonist, there isn’t any contradiction at all in the increasingly smaller line segments scenario. He doesn’t need to believe that these line segments actually exist in some ‘reality’, and he doesn’t need to believe that a sum of all a limitless quantity of line segments ‘exists’. All he does is to observe that the only statements that he can make about these line segments are logical statements regarding the definitions involved, some of which involve limitlessness. And he observes that the process of continually making further line segments never actually finishes, and so a point is never reached where there is a contradiction. For the non-Platonist, the limit of the sums of such fractions as 1⁄10 + 1⁄100 + 1⁄1000 + 1⁄10000 + 1⁄100000 + … is 1⁄9, but he also acknowledges that that limit is never actually reached. (Footnote: Historically, there have been intense arguments about whether a limitlessness could be considered as completed.)
But while the Courant & Robbins contradiction isn’t a contradiction for the non-Platonist, and doesn’t present any difficulties for the non-Platonist, for the Platonist it is an entirely different matter. He cannot state that the sum of a limitless quantity of mathematical entities does not ‘exist’ – because it is a cornerstone of Platonist philosophy that sums of limitless quantities of mathematical entities do ‘exist’, and moreover, that they ‘exist’ independently of any human definition. (Footnote: For example Platonists believe that there are real numbers that are the sum of a limitless number of fractions and which can never be defined by any finite means – and that means that they can only ‘exist’ as ‘actual’ Platonist things.)
So, from the point of view of the Platonist:
and the combination of these beliefs results in a contradiction.
A line segment is simply a part of a line that defines a set of all numbers greater than (or greater than or equal to) a given value and less than (or less than or equal to) a given value, so, since the Platonist insists that the sum of an infinite quantity of fractions does ‘exist’, he cannot logically argue that such line segments don’t ‘exist’. Similarly, it would be absurd for a Platonist to argue that while the sum of an infinite quantity of fractions does ‘exist’, the sum of an infinite quantity of line segments cannot ‘exist’.
The Courant & Robbins contradiction is a compelling argument against Platonism in mathematics. It demonstrates that the Platonist belief that the sum of a limitless quantity of fractions such as 1⁄10 + 1⁄100 + 1⁄1000 + … ‘exists’, and is the same ‘actual’ thing as the value we refer to as ‘1⁄9’, is logically untenable.
For the non-Platonist, every one of these fractions 1⁄10, 1⁄100, 1⁄1000, 1⁄10000 … is a finite size, however small it may be; and all the fractions are interrelated, since every fraction is 1⁄10 of a previous fraction. And every fraction is related to the first fraction, which is 1⁄10 by being multiplied by 1⁄10 a finite number of times, that is, every such fraction can be expressed as (1⁄10)n where n is some natural number. That means that all such fractions, by the very definition of the series, are of finite size; there cannot be any one of these fractions that is not a definite determined size. But the Platonist view that all such fractions ‘exist’ requires that there ‘exists’ more than a finite quantity of such fractions – and that requires that there must be fractions that are related to the first fraction (which is 1⁄10) by being multiplied by 1⁄10 a limitless number of times. But the Platonist cannot give any value for 1⁄10 multiplied by itself a limitless number of times. He cannot give it any finite value, and neither can he give it a value of zero – because, according to the Platonist creed, there ‘exist’ in some reality, limitlessly many such fractions which consist of 1⁄10 multiplied by itself a limitless number of times, so they can’t all be the value of zero. Of course, the non-Platonist can say that the limiting value as you repeatedly multiply 1⁄10 by 1⁄10 over and over again is zero, that is, the value of (1⁄10)n approaches zero as the value of n gets bigger and bigger – but it never reaches that value. But that is of no help to the Platonist when faced with the Courant & Robbins contradiction. He has no way of circumventing the absurdity of the Platonist viewpoint that the sum of more than a finite quantity of such fractions ‘exists’.
For more demonstrations of contradictions arising from the Platonist beliefs in the ‘existence’ of ‘actual’ infinite sets, see Sums of infinitely many fractions: 1 and Sums of infinitely many fractions: 2.
The page Lebesgue measure theory demonstrates in more detail why the conventional assumption that you can add the lengths of infinitely many intervals as an infinite sum such as 1⁄10 + 1⁄100 + 1⁄1000 + … is naive and simplistic, and overlooks a crucial fact.
You can also see a formal paper on some of the problems of calculating the total measure of some sets that are defined in terms of limitlessness, see On Smith-Volterra-Cantor sets and their measure (PDF).
For a limitlessly long geometric series in the form of
the sum of the first n + 1 terms is:
Provided the value of r is greater than 1, then the value of decreases as n increases, and the limiting value of as n increases is zero.
That gives the standard result that the limiting value of the sum as n increases, provided r is greater than 1 is:
which is the same as
For the series 1⁄10 + 1⁄100 + 1⁄1000 + 1⁄10000 + 1⁄100000 … r is 10, and so the limit of the sum as n increases is given as
which evaluates as 1⁄9. And that means that, no matter how many terms are in the series, the value of the series can never exceed the value 1⁄9. To put it another way, the limiting value that any such sum of a finite quantity of terms approaches, but never actually reaches, is 1⁄9. Note that the quantity
in the formula can never actually be zero; every fraction that is in the series is of the form
where y is always some finite number. There is no fraction in the series where y is not some finite number.
Note that 0.999… is usually used to indicate the limiting value of the series:
9⁄10 + 9⁄100 + 9⁄1000 + 9⁄10000 + 9⁄100000 …
Since this is the same as 9 × (1⁄10 + 1⁄100 + 1⁄1000 + 1⁄10000 + 1⁄100000 …), then the limiting value is 9 × 1⁄9, which is 1.
There is more than one way of making a list of the rational numbers. An easy way is demonstrate is to visualize a method, by using horizontal rows, the first with all fractions with 1 on the top, the next row with all the fractions with 2 on the top, and so on, like this:
Then we can envisage a simple way of progressing through the fractions so as to not miss any on the way. We can do this by taking each column in turn, like this:
This scheme outlines a way of listing all the positive rational numbers. Although it is a visual scheme, there is no reason why the methodology could not be defined without using any visual cues. Of course, the list never finishes, and that is because there is no limit to the quantity of rational numbers. You might notice that in this scheme some numbers are repeated, since 1⁄1 = 2⁄2, and 2⁄1 = 4⁄2. This is true, but the important point is that the scheme doesn’t omit any rational numbers. And if you wanted to include negative numbers, then you do the same scheme except that you simply include a negative of each number immediately after each number.
If you want to list only the fractions between say, 0 and 1, then you simply leave out the fractions that are greater than one, like this:
and make a list in the same way as before.
Some people have suggested that they can circumvent the contradiction by using a list (see also One-to-one correspondences and Listing the rationals) of the rationals that are defined in terms of various conditional requirements, which render the list and the sequence of intervals interdependent. Rather than trying to construct a set of rules as to which lists are applicable, all that is required is one specific list. We can define that the set A is to be given by one specific list using the pattern of rationals:
We go through this pattern, leaving out any duplicates, which gives the first terms of the list as
1⁄2, 1⁄3, 2⁄3, 1⁄4, 3⁄4,1⁄5, 2⁄5, 3⁄5, 4⁄5, 1⁄6, 5⁄6, …
Given this list, there are no points in the interval 0 to 1 that are not in the set A.
Note that this list follows a pattern that for each subsequent denominator, the values run from the lowest to the highest value of the numerator. For every subsequent denominator, this gives a pattern of rationals across the interval 0 to 1 which is mirrored about 1⁄2. This patterning continues infinitely as the terms progress.
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There is now a new page on a contradiction in Lebesgue measure theory.
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