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.

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

A Simplified Explanation of Gödel’s proof - Part 9


 

 


 

Part 9: The Flaw In The Proof

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:

 


 

 


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 …  

 

The Lighter Side

 

NEWS

Lebesgue Measure

There is now a new page on Lebesgue measure theory and how it is contradictory.

 

 

Illogical Assumptions

There is now a new page Halbach and Zhang’s Yablo without Gödel which demonstrates the illogical assumptions used by Halbach and Zhang.

 

 

Peter Smith’s ‘Proof’

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.

 

 

Easy Footnotes

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).

 

 

O’Connor’s “computer checked” proof

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.

 

 

New page on Chaitin’s Constant

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.

 

Previous Blog Posts  

 

13 May 2015 Good Math, Bad Math?

 

30 Apr 2015 The Chinese Room

 

31 Mar 2015 Cranks and Crackpots

 

16th Mar 2015 Bishops Dancing with Pixies?

 

23rd Feb 2015 Artificial Intelligence

 

Links  

 

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:

Gödel Links

 

– and a page relating specifically to the Gödel mind-machine debate:

Gödel, Minds, and Machines

 

Printer Friendly

 

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

 

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