Logic and Language

Logic and Language

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

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In Part 7: Another Numbering System the Basic Numbering function, which we called BN, was explained. Gödel defined the Basic Numbering function so that it would always give the same value as the Gödel Numbering function. When he did that, he made a proposition:

*‘For any number, the Basic Number of that number is the same as the Gödel Number of that number’*

We can also state that as:

*‘For every x, where x is a number, BN*

It looks innocent enough, but let’s look a bit deeper - BN and GN are defined as relationships. But what language do they belong to?

Let’s look at the Gödel Numbering function, ** GN(x)**. The values that the free variable x can take are symbols of the formal language – and combinations of symbols of the formal language. That means that

Now look at the Basic Numbering function, ** BN(x)**. Gödel uses this function to get his ‘true but unprovable’ sentence. And again, it’s no coincidence that it also occurs in the ‘true but unprovable’ sentence in our simplified proof.

That sentence was:

*‘There is no G‑proof of G‑Substitution *

For the proof to work, that sentence has to be a number relationship. That means that since BN occurs in this sentence BN also has to be a number relationship. And that means that the variable x in ** BN(x)** is a variable of a number relationship.

Now, Gödel has to use the proposition: *‘For every x, BN*

But that proposition is nonsensical because it mixes up the meta-language and its sub-languages. In that expression the variable x is at the same time a variable of the meta-language and a variable of a sub-language – the language of number relationships. That is a contradiction that shows that Gödel’s proof is wrong.

In the Gödel Numbering function, x has to be a variable of the meta-language – but in the Basic Numbering function, x has to be a variable of a number relationship. And that means that the expression:

*‘For every x, BN*

isn’t actually a proposition at all. It is absurd since it doesn’t follow the basic rules for a valid proposition.

And if we use symbols for the variables of the meta-language that are different to the symbols for variable of the language of number relationships, then we can’t even write the expression

*‘For every x, BN*

since the variable in ** BN( )** can’t be x. And if we can’t even write the expression as a logical expression, it cannot be used to prove anything. And that is why Gödel’s proof is wrong.

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?.

How you can tell if someone is a crackpot

A review of Buldt’s *The Scope of Gödel’s First Incompleteness Theorem*

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.

8 Apr 2016 Are we alone in the Universe?

13 May 2015 Good Math, Bad Math?

31 Mar 2015 Cranks and Crackpots

16th Mar 2015 Bishops Dancing with Pixies?

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

www.jamesrmeyer.com