Logic and Language

Logic and Language

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

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Richard’s paradox is as follows: (Footnote: Richard, Jules (1905). *Les Principes des Mathématiques et le Problème des Ensembles* (The principles of mathematics
and the problem of sets).)

English consists of various symbols - 26 letters, spaces, and various punctuation marks.

All permutations of these symbols of the English alphabet can be put in a list, by first taking all permutations of two symbols in an alphabetical order, then taking all permutations of three symbols in order, and continue the list by taking all permutations of four symbols, and so on. The permutations may contain the same symbol repeated several times.

For any integer ** n**, any permutation of the symbols taken

Given this list, we can now define a number **N** not belonging to this
list:

“Let * p* be the digit in the

This number **N** does not belong to the list. If it were the ** n^{th}** number in the list, the digit at its

Richard actually goes on to say that there is only an * apparent contradiction*, as follows:

“*The collection of symbols within the quotes, which we will call G, is a permutation that will appear in the list E. But it is defined in terms of the set E, which
has not yet been defined. Hence it will not appear in the list E. G has a defined meaning
only if the list E is totally defined*.”

Here Richard hits the nail on the head, and points out that the supposed paradox is dependent on the ambiguity of the definitions involved. But it seems that almost everyone prefers to ignore this fact and instead believe that there is something significant in it.

And if you try to actually provide a clear definition of the list, you will need to clarify the vague definition above. The list is defined in terms of a list **B** that includes * every* English expression, then if that list

Since the list **B** cannot be defined within the English language, then the list **E** of all English expressions for natural numbers cannot be defined in English either, since the list **E** of all English expressions for natural numbers is defined in terms of the list **B**. Hence the argument that there is a number **N** that is not in the list **E **fails, since it assumes that the definition of **N** is possible as an English expression - but that is not possible, since it is defined in terms of the list **E**, which is defined in terms of the list **B**, which cannot be defined in English.

This is all blindingly obvious if one at all familiar with the notions of sub-language and meta-language, since the list that can list all symbol sequences of a given language must be in a language that is a meta-language to that given language, and cannot be in the given language itself.

A variation of the paradox uses natural numbers instead of real numbers, and supposes a list of all English expressions that describe properties of natural numbers.

It relies on the claim:

An expression that is the ** n^{th}** expression in the list can describe a property that may or may not apply to the

and from that claim the assertion:

If the ** n^{th}** expression describes a property that does

The paradox arises as follows:

Since the property of being Richardian is itself a numerical property of integers, it must be included in the list of all definitions of properties. Therefore, the property of being Richardian is the ** n^{th}** expression in the list for some number

Is * n* Richardian? If

But again, as in the case of the assumption of a list of English expressions for real numbers, the definition of the list itself cannot be in the same language as the list, and hence there cannot be an expression in the list that can refer to the definition of the list. Hence the supposed Richardian expression cannot occur in the list, and the supposed paradox disappears.

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 new page on a contradiction in Lebesgue measure theory.

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.

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