Logic and Language

Logic and Language

Copyright © James R Meyer 2012 - 2018 www.jamesrmeyer.com

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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:

- Suppose that there can be a list (for definition of what is meant by a list, see one-to-one correspondence) that includes all of the real numbers. For our purposes, we will only consider the real numbers between
**0**and**1**. For convenience we will use the concept of the real number line between**0**and**1**, and which represents all the real numbers between**0**and**1**. The ‘proof’ operates by enclosing every real number in the list between**0**and**1**by a small length of a line as follows:- For the first real number in the supposed list of all real numbers, we make a line segment that is
wide, and which includes that real number (note that the number can be anywhere within this line segment).^{1}⁄_{10} - For the second real number in the list, make a line segment
wide, enclosing that real number.^{1}⁄_{100} - For the third real number in the list, make a line segment
wide, enclosing that real number.^{1}⁄_{1000} - And so on, through the list, for all the real numbers. As long as the appropriate real number is within the correct line segment, it is immaterial precisely where in that line segment it lies.

- For the first real number in the supposed list of all real numbers, we make a line segment that is
- The proof says that if every real number between
**0**and**1**is actually in the list, then the entire line between**0**and**1**must be covered by this infinite series of line segments whose lengths get progressively smaller:^{1}⁄_{10,}^{1}⁄_{100,}^{1}⁄_{1000,}^{1}⁄_{10000,}…^{1}⁄_{100000} - Of course, some of the line smaller segments may overlap. But, says the proof, even if there were no overlapping at all, the actual total length of all our line segments is the sum of all the lengths

+^{1}⁄_{10}+^{1}⁄_{100}+^{1}⁄_{1000}+^{1}⁄_{10000}…^{1}⁄_{100000} - The proof says that the sum of all these lengths is
, because^{1}⁄_{9}+^{1}⁄_{10}+^{1}⁄_{100}+ … is what is called a standard geometric series, and the proof says that it is a standard result that the sum of such a series as^{1}⁄_{1000}(See^{1}⁄_{9}*Appendix: Geometric Series*). - The proof continues: Since, even if there is overlapping, the sum of the segments cannot be greater than
, and that since that is less than the total length of the line segment from^{1}⁄_{9}**0**to**1**(which is a length of**1**), then we cannot actually have included all the real numbers in the line segments that we used. - Therefore, the ‘proof’ continues, the original supposition that could be a complete listing of the real numbers must have been incorrect, and there cannot be a listing of all the real numbers. Because of that, there must be more real numbers than natural numbers.

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

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

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

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

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

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

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

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:

- a limitless quantity of line segments such as
^{1}⁄_{10,}^{1}⁄_{100,}^{1}⁄_{1000,}… actually ‘exists’, and^{1}⁄_{10000} - the sum of such a limitless quantity of line segments ‘exists’ and the value of that sum is
, and^{1}⁄_{9} - the pairing of every rational number (between 0 and 1) and some line segment such as
^{1}⁄_{10,}^{1}⁄_{100,}^{1}⁄_{1000,}… ‘exists’,^{1}⁄_{10000}

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 ‘

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

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).

Footnotes:

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

That gives the standard result that the * limiting* value of the sum as

which is the same as

For the series ** ^{1}⁄_{10}** +

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

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}** +

Since this is the same as **9 ×** (** ^{1}⁄_{10}** +

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}** =

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:

^{1}⁄_{2} |
^{1}⁄_{3} |
^{1}⁄_{4} |
^{1}⁄_{5} |
^{1}⁄_{6} |
… |

^{2}⁄_{3} |
^{2}⁄_{4} |
^{2}⁄_{5} |
^{2}⁄_{6} |
… | |

^{3}⁄_{4} |
^{3}⁄_{5} |
^{3}⁄_{6} |
… | ||

^{4}⁄_{5} |
^{4}⁄_{6} |
… | |||

^{5}⁄_{6} |
… |

We go through this pattern, leaving out any duplicates, which gives the first terms of the list as

** ^{1}⁄_{2}**,

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 Halbach and Zhang’s *Yablo without Gödel* which analyzes the illogical assumptions used by Halbach and Zhang.

I found that making, adding or deleting footnotes in the traditional manner proved to be a major pain. So I developed a different system for footnotes which makes inserting or changing footnotes a doddle. You can check it out at Easy Footnotes for Web Pages (Accessibility friendly).

I have now added a new section to my paper on Russell O’Connor’s claim of a computer verified incompleteness proof. This shows that the flaw in the proof arises from a reliance on definitions that include unacceptable assumptions - assumptions that are not actually checked by the computer code. See also the new page Representability.

There is now a new page on Chaitin’s Constant (Chaitin’s Omega), which demonstrates that Chaitin has failed to prove that it is actually algorithmically irreducible.

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13 May 2015 Good Math, Bad Math?

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For convenience, there are now two pages on this site with links to various material relating to Gödel and the Incompleteness Theorem

– a page with general links:

– and a page relating specifically to the Gödel mind-machine debate:

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