Logic and Language

Logic and Language

Copyright © James R Meyer 2012 - 2017 www.jamesrmeyer.com

This page is keyboard accessible:

• Use**Tab**, **Shift + Tab **keys to traverse the main menu. To enter a sub-menu use the **Right Arrow** key. To leave a sub-menu use the **Left Arrow** or the **Escape** key.

• The**Enter** or the **Space** key opens the active menu item.

• To skip the menu and move to the main content, press**Tab** after the page loads to reveal a skip button.

• To get back to the top of the page anytime, press the**Home** key.

• For more information, click here: Accessibility Close this tip.

• Use

• The

• To skip the menu and move to the main content, press

• To get back to the top of the page anytime, press the

• For more information, click here: Accessibility Close this tip.

Note: Full functionality of this web page requires JavaScript to be enabled in your browser.

A favorite notion held dear by Platonists is the notion of an ‘infinite decimal expansion’. For a fraction, the decimal representation of the fraction is simply a fraction with 10, 100, 1000, 10000, etc as the denominator (the denominator is the number under the line in a fraction), so that we can have, for example ** ^{9}⁄_{10}**,

Of course, when trying to calculate the value of the square root of 2 (the value that, when multiplied by itself gives the value of 2) we can continue to generate more and more fractions (or decimal places), viz.

1.4

1.41,

1.414,

1.4142,

1.41421,

…

…

1.4142135623730950488016887242097,

…

This process of generating more and more decimal places can be continued indefinitely. There is no limit to the number of times you can add on an extra digit.

Based on this simple fact a crucial assumption was made, an assumption which is an inherently Platonist assumption. That assumption is that the irrational number is a thing that actually consists of a more than a finite number of decimal digits. For the Platonist, the value of the square root of **2** is actually equal to:

**1** plus the ‘fraction’:

414213562 …
to more than a finite number of digits1000000000 …
to more than a finite number of digits |

When confronted with this, the Platonist will reply that that is not what they mean by an ‘infinite decimal expansion’, and that what they really mean is the sum of a limitless quantity of fractions that start off as:

** ^{4}⁄_{10} + ^{1}⁄_{100} + ^{4}⁄_{1000} + ^{2}⁄_{10000} + ^{1}⁄_{100000} + ^{3}⁄_{1000000} + ^{5}⁄_{10000000} + ^{6}⁄_{100000000} + ^{2}⁄_{1000000000} +** …

and that there are limitlessly many such fractions that continue to get smaller and smaller, all added together.

But this doesn’t solve the problem of a limitlessly large denominator; it is simply evading the problem by a rewording rather than a different definition. Each fraction in this ‘sum’ is a definite size and its denominator is ** ^{1}⁄_{10}** of the previous one. But if this is a limitless ‘sum’ that can actually ‘exist’, then there must actually be more than a finite quantity of fractions, each one of a definite size and greater than zero, all added together. From this arises a conundrum:

“*Every fraction is related to every other fraction in the ‘sum’, since the value of its denominator must be the value of the denominator of the first fraction in the sum, which is 10, multiplied by 10 a definite number of times. So, if there could be more than a finite number of these fractions, then there would be some fraction that was not related to any of the other fractions by any finite multiple of ten. But that is impossible, since every such fraction has to be related to every other fraction by some finite multiple of 10.*”

At this point, the Platonist will resort to virtual hand-waving, and will say something like:

“*But you have presumed that every fraction is a finite amount away from the first fraction. But some of the fractions will be an infinite amount away from the first fraction, and so your argument doesn’t apply.*”

Unfortunately for the Platonist, this attempt to solve the conundrum doesn’t work. Every one of the fractions has to have a denominator of a definite size, that is, the denominator must be a **1** followed by a definite number of zeros. The alternative is that there are fractions in this infinite sum that do not have a definite value, and that their denominator is **1** followed by an infinite number of zeros – which contradicts the original definition.

Unlike the notion of the sum of an infinite quantity of ever decreasing fractions, a concept that *does* have a clear meaning is the *limit* of all the *finite* sums that can be got from a collection of fractions. For instance, given a collection of fractions that is:

** ^{4}⁄_{10}** ,

you can have sums such as:

^{4}⁄_{10} + ^{1}⁄_{100}

^{4}⁄_{10} + ^{1}⁄_{100} + ^{4}⁄_{1000}

^{4}⁄_{10} + ^{1}⁄_{100} + ^{4}⁄_{1000} + ^{2}⁄_{10000} + ^{1}⁄_{100000}

^{4}⁄_{10} + ^{1}⁄_{100} + ^{4}⁄_{1000} + ^{6}⁄_{100000000} + ^{2}⁄_{1000000000}

^{4}⁄_{10} + ^{4}⁄_{1000} + ^{2}⁄_{10000} + ^{1}⁄_{100000} + ^{5}⁄_{10000000} + ^{2}⁄_{1000000000}

^{4}⁄_{10} + ^{1}⁄_{100} + ^{4}⁄_{1000} + ^{2}⁄_{10000} + ^{1}⁄_{100000} + ^{3}⁄_{1000000} + ^{5}⁄_{10000000}

^{4}⁄_{10} + ^{1}⁄_{100} + ^{4}⁄_{1000} + ^{2}⁄_{10000} + ^{1}⁄_{100000} + ^{3}⁄_{1000000} + ^{5}⁄_{10000000} + ^{6}⁄_{100000000}

^{4}⁄_{10} + ^{1}⁄_{100} + ^{4}⁄_{1000} + ^{2}⁄_{10000} + ^{1}⁄_{100000} + ^{3}⁄_{1000000} + ^{5}⁄_{10000000} + ^{6}⁄_{100000000} + ^{2}⁄_{1000000000}

All of these sums are finite. None of them exceed a certain value. None of them exceed the value 0.414213562. That value is the limiting value for finite sums of fractions taken from the finite collection:

** ^{4}⁄_{10}** ,

And if you have more fractions to choose from, you will have many more different finite sums of fractions. The above was a finite collection of fractions from which there can only be a finite number of different sums. But you can also have a concept of a limiting value for all the finite sums that can be got from a limitlessly large collection of fractions. In this case, there are a limitless amount of such sums – but the sums are still all finite. Since all these finite sums are derived from a concept of a collection of fractions that has limitlessly many fractions, then, in order to be able to be able to determine if there is a limit which none of these finite sums can exceed, there must first of all be a way of expressing that concept of a collection of limitlessly many fractions in a finite way, using a finite combination of symbols. Since the concept involves a limitless amount of such fractions, the only way to express that concept is by way of a definition. And similarly, once you have that finite definition, you may be able to use that finite definition to define a limiting value. And that limiting value will be a value that no finite sum of fractions, taken from the overall collection that is given by a definition, can exceed. None of this is anything new; most of the methods of calculating the limits for various series of fractions were known over a hundred and fifty years ago.

Of course, we can have numbers which are finite decimal expansions, such as

3.1416, and

3.14159265359, and

3.14159265358979323846, and

3.14159265358979323846264338328, and 3.141592653589793238462643383279502884197169399,

and so on, which are all numbers that are close to the number **Pi**, *relative* to the number **3**. And a number has been calculated to billions of digits and this is also closer to the number **Pi**, *relative* to the number 3.14159265358979323846. But, in the same way as it is a mistake to think of a ‘large’ number as being close to infinity, it is a mistake to think that the number 3.141592653 is close in some absolute sense to the value of **Pi**. It isn’t, since there are still infinitely many real numbers between 3.141592653 and **Pi.** It is only close in ** relative** terms, in that it is closer to

For more demonstrations of contradictions arising from the Platonist beliefs in the ‘existence’ of ‘actual’ infinite sets, see The Courant & Robbins contradiction and Sums of infinitely many fractions: 1.

Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked (comments will be checked before appearing on this site). Difficulties in understanding the site content are usually best addressed by contacting me by e-mail. Note: you will be asked to provide an e-mail address - this will only be used to notify you of replies to your comments - it will never be used for any other purpose, will never be displayed and does not require verification. Comments are common to the entire website, so please indicate what section of the site you are commenting on.

If you cannot see any comments below, it may be that a plug-in on your browser is blocking Disqus comments from loading. Avast anti-virus in particular is known to do this, especially with Internet Explorer and Safari. See Disqus Browser plug-in/extension conflicts or Why isn’t the comment box loading?.

Please wait for comments to load …

It has come to my notice that, when asked about the demonstration of the flaw in his proof (see A Fundamental Flaw in an Incompleteness Proof by Peter Smith PDF), Smith refuses to engage in any logical discussion, and instead attempts to deflect attention away from any such discussion. If any other reader has tried to engage with Smith regarding my demonstration of the flaw, I would be interested to know what the outcome was.

There is a new addition to the page Yet another flawed incompleteness proof, where Berto’s proof of incompleteness in his book *There’s something about Gödel* comes under scrutiny.

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.

13 May 2015 Good Math, Bad Math?

16th Mar 2015 Bishops Dancing with Pixies?

23rd Feb 2015 Artificial Intelligence

31 Mar 2015 Cranks and Crackpots

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:

All pages on this website are printer friendly, and will print the main content in a convenient format. Note that the margins are set by your browser print settings.

Note: for some browsers JavaScript must be enabled for this to operate correctly.

Comments on this site are welcome, please see the comment section.

Please note that this web site, like any other is a collection of various statements. Not all of this web site is intended to be factual. Some of it is personal opinion or interpretation.

If you prefer to ask me directly about the material on this site, please send me an e-mail with your query, and I will attempt to reply promptly.

Feedback about site design would also be appreciated so that I can improve the site.

Copyright © James R Meyer 2012 - 2017

www.jamesrmeyer.com