Logic and Language

Logic and Language

Copyright © James R Meyer 2012 - 2016 www.jamesrmeyer.com

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The key point of Gödel’s paper is the part that he calls his Fifth Proposition. In fact, it isn’t actually his fifth proposition; what he really meant was that it was his fifth key proposition. The entire result of his theorem depends on this Fifth Proposition. Since it’s the crux of the proof, you might expect that he would go into it in great detail, and set out the proof of this proposition in a very clear and logical fashion. However, he only provides a brief outline of how you might create a proof. (Footnote: Peter Smith, although he is a staunch advocate of Gödel’s proof, acknowledges in his paper, *Expounding the First Incompleteness Theorem (PDF)*, that, *“Gödel only sketches a proof… The crucial step is just asserted.”*) Now, there’s nothing wrong with having a new idea, and writing down a brief synopsis of your idea. But to rely on that brief synopsis instead of giving a rigorous and complete proof is simply not good enough. To assume a theorem is correct because you have a gut feeling that it’s correct simply isn’t acceptable.

To show you why Gödel’s assumption that his Fifth Proposition is correct is completely wrong, we’ll look at the simplified version of the proof that has just been demonstrated to you, because the principles are the same. The principles are the principles involved in dealing with meta-languages and sub-languages. And Gödel’s proof has to be stated in a language that is a meta-language to the formal language.

I’m going to recap the differences between a variable and the specific values that it represents, because a proper understanding of this simple concept is crucial to understanding the flaw in Gödel’s theorem.

When I say the proposition:

‘Every person has a heart’

I’m implying the proposition:

‘James R Meyer has a heart’.

All that we did there was to substitute the variable ‘person’ by the value ‘James R Meyer’, and I removed the quantifier ‘Every’ and I got the new expression, ‘James R Meyer has a heart’. And when I refer to ‘James R Meyer’ in this expression, ‘James R Meyer’ is obviously not a variable, but a specific person (me).

And that simply shows that whenever there’s a word which is a variable of a language, any values that it represents aren’t variables – the values that it represents are specific values in that language. But when you have meta-languages and sub-languages, a variable in one language can be a specific value in another language. And when that is the case, you have to be very careful that you don’t mix up your languages.

Now we’re ready to talk about number relationships and sub-language.

In Part 6: Matching number relationships and formal sentences, we came across a proposition. That proposition was:

*‘For any number relationship with one free variable, there is a matching formal sentence with one free variable, where that formal sentence expresses the same concept as that number relationship.’*

Since ‘number relationship’ is quantified by ‘For every’ that means that ‘number relationship’ is a variable in the language of that proposition. That language is Gödel’s proof language – his meta-language. That means that the specific values to which the variable ‘number relationship’ refers to are specific number relationships. And that means that in the language of Gödel’s proof, any specific number relationship is a specific value.

It’s no coincidence that Gödel’s Fifth Proposition also states essentially the same thing as the proposition in this simplified proof, which was:

*‘For any number relationship with one free variable, there is a matching formal sentence with one free variable, where that formal sentence expresses the same concept as that number relationship.’*

And the word ‘variable’ in that proposition is itself a variable in the language of that proposition. The specific values that it refers to are symbols that are variables of number relationships. That means that any symbol that is a variable of a number relationship is a specific value in the language of Gödel’s proof.

So we can say that in the language of Gödel’s proof, number relationships are specific values, and variables of number relationships are also specific values. That means the language of Gödel’s proof has to be a language that is a meta-language to number relationships, and that whatever language a number relationship might be expressed in, that is a sub-language to that meta-language.

Now, remember, Gödel’s proof language is a meta-language to the formal language. So in that meta-language, formal sentences are specific values, and the variables of formal sentences are specific values.

And that means that the language of Gödel’s proof is a meta-language to both the formal language and to the language of his number relationships. His meta-language is a meta-language that talks about formal sentences and talks about number relationships.

The fundamental flaw in Gödel’s proof is that he confuses his meta-language and the language of number relationships. Although the language of number relationships is a sub-language, Gödel makes the mistake of assuming that number relationships can also actually be valid expressions of his meta-language – but they can’t – because of his statement of his Fifth Proposition.

Since the formal language is a sub-language, that means that formal sentences aren’t expressions of the meta-language. They’re simply combinations of symbols that the meta-language talks about. And exactly the same thing applies to number relationships. Number relationships can’t be expressions in that meta-language either – they’re simply combinations of symbols that the meta-language talks about. That means that neither number relationships nor formal sentences are expressions in the meta-language – they’re seen by the meta-language simply as combinations of symbols. And as already mentioned, that means that a number relationship has no meaning in the meta-language. It’s just a combination of symbols that is a specific value in the meta-language.

In order that Gödel’s proof language can translate a number relationship into a formal sentence, it has to use variables to refer in general terms to the symbols that make up the formal language and which make up number relationships. Because it uses its own variables to refer to those values, any value that is one of those specific values can’t be a variable of the meta-language. That means that the variables of formal sentences can’t be variables of the meta-language; and neither can the variables of number relationships.

As I said, in his proof, Gödel didn’t bother actually providing a proper proof of his Fifth Proposition. All he did was to provide a brief outline. But in fact his Fifth Proposition is completely wrong. And the reason it’s completely wrong is that Gödel fails to understand that when he talks about number relationships and variables of number relationships, the language he’s using has to be a logically valid language. And that means it has to be a meta-language to the language of number relationships.

Once you realise that Gödel’s proof language is a meta-language to the language of number relationships as well as to the formal system, there are a variety of ways that you can show that Gödel’s proof cannot possibly give the result that Gödel intended (if it is to be a logical proof).

Footnotes:

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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.

Flawed proofs of the Diagonal Lemma by Panu Raatikainen and Vann McGee have been added to the Diagonal Lemma web page.

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:

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Copyright © James R Meyer 2012 - 2016

www.jamesrmeyer.com