Logic and Language

Logic and Language

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

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Cantor published a proof in 1874 that the real numbers are not denumerable, that is, that there cannot be a list that can match every one of the real numbers uniquely to every one of the natural numbers. (Footnote: Georg Cantor,‘*Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen*’, Journal für die Reine und Angewandte Mathematik 77 (1874), pp 258-62, English translation online at Cantor’s 1874 Proof of Non-Denumerability - English Translation. Cantor also published a similar proof in 1884: ‘*Über unendliche lineare Punktmannigfaltigkeiten*’, Mathematische Annalen 6, 23 (1884) pp 453-488.) There is a English translation of Cantor’s original proof online at Cantor’s 1874 Proof of Non-Denumerability - English Translation.

In mathematical terms, the proof claims that there can be no mathematical function that will map every natural number to some real number so that every natural number has a corresponding real number. This proof is essentially as follows:

First of all we assume that there actually can be a function matching the natural numbers to all the real numbers; the object of the proof is to prove that this assumption cannot be correct. We call the assumed function ** List(x)**. Whenever we replace the

Now we go through the natural numbers one by one, starting at zero, until we get two values for the function ** List(x)** that are between the values 0 and 1 (which will be real numbers) – we call these first two natural numbers

We repeat this procedure and continue to go through the natural numbers one by one until we find the next two values of ** List(x)** (again, these are real numbers) that are within the interval

We continue this process again and again. As we do so, the interval becomes smaller and smaller. But between any two real numbers there is always another real number, since there is no such thing as a smallest difference between two real numbers. That means that no matter how long we continue the process for, there will still be two real numbers with an interval between them.

The proof now says: Seeing as the interval never vanishes, there must be at least one number (which we will call ** c**) which lies between every interval of every

This is a contradiction, since it must lie between the sequence ** List(n_{1,1,1, …})** and the sequence

The above proof proves, of course, that there cannot be a function that maps the natural numbers to the real numbers, where the function is in the same language as the real numbers.

We note that the proof makes no mention of levels of language - no mention of meta-language or sub-language - and makes no assertions about functions that map natural numbers to real numbers but are not in the same languages as those real numbers (i.e., such function are in a meta-language to the real numbers of the list.

On The Diagonal Proof page, it is shown that if levels of language are ignored and unfounded Platonist assumptions are included, then, by the secondary argument of the Diagonal proof, this generates the result is that there are ‘more’ real numbers than natural numbers. See also the paper On Considerations of Language in the Diagonal Proof (PDF).

In this respect, the only difference between the above proof and the Diagonal proof is the difference in the description of the Diagonal number and the number ** c**. So, in the above proof, if it is assumed that the number

As for the Diagonal argument, if we do not include the implicit Platonic assumptions that support the secondary argument, then the proof simply proves that there cannot be a matching function ** List(x)** from all natural numbers to all real numbers,

And as for the Diagonal argument, if the function ** List(x)** is a function is in a meta-language, then we cannot simply assume that there can be a valid mathematical definition of

As in the case of the Diagonal argument, without unfounded assumptions concerning the ‘existence’ of things independently of language, Cantor’s first proof proves that there cannot be any function that maps the natural numbers to the real numbers, where the function is in the same language as the real numbers - but it provides no basis for the secondary argument, that there ‘exist’ undefinable real numbers.

For more details, see the Diagonal proof, the secondary argument of the Diagonal proof, and A List with no Diagonal number.

Footnotes:

Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked. 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 - any address will do, it does not require verification. Your e-mail will only be used to notify you of replies to your comments - it will never be used for any other purpose and will not be displayed. If you cannot see any comments below, see Why isn’t the comment box loading?.

There is now a paper that deals with the matter of language and the diagonal proof, see On Considerations of Language in the Diagonal Proof.

There is now a new page on a contradiction in Lebesgue measure theory.

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.

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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Copyright © James R Meyer 2012 - 2018

https://www.jamesrmeyer.com