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A Simplified Explanation of Gödel’s proof - Part 6


 

 


 

Part 6: Matching number relationships and formal sentences

We now get to the point in the proof where we assert:

 

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.

 

A relationship that’s a purely number relationship is simply a relationship between numbers that is expressed in some language that is not the formal language. For example, there can be many different ways of expressing the concept of x plus x is equal to six’. We could say x and x add up to six’, or we could say ‘two times x adds up to six’. But as long as it is clearly defined, then it could always be translated into the formal language.

 

And we know that for any formal sentence, there’s a matching Gödel number.

 

That means that, given a number relationship with one free variable, since there’s a matching formal sentence with one free variable, and since that formal sentence has a matching Gödel number, that number is also a matching Gödel number for that number relationship.

 

As a simple example, we take the number relationship x plus x is equal to six’. Let’s assume that in our formal language the matching formal sentence is:

 

{x + x = ffffff0}

 

where in the formal system 0 is zero, f0 is one, ff0 is two, fff0 is three, and so on.

 

Purely for the purposes of demonstration we assume that the matching Gödel number for that formal sentence is 12743 (although it would actually be a much bigger number). That means the matching Gödel number for x plus x is equal to six’ is also 12743.

 

Now, a number relationship with a free variable isn’t a proposition. But if the variable is substituted by some specific value, it can be a proposition. So if we substitute the free variable of our number relationship by some number, and we substitute the free variable of the formal sentence by the same number, then we can have a number relationship that is a proposition about numbers that has a matching formal sentence that is a sentence about numbers.

 

For our example, if we substituted the x by three, we would get ‘three plus three is equal to six’, and the matching formal sentence would be {fff0 + fff0 = ffffff0}.

 

 

Provability and truth

Now that we have a matching formal sentence and a number relationship, then we can say that:

 

‘If a formal sentence is provable, then the matching number relationship must be true.’

 

It’s pretty obvious that that has to be the case – provided that the formal language is consistent (that means it isn’t able to prove false sentences).

 

So, with our previous example, if the formal language can prove that {fff0 + fff0 = ffffff0}, then it has to be true that three plus three is equal to six.

 

And following on from the above, we can now say that:

 

‘For any sentence of the formal language, either there’s a formal proof of that sentence or there isn’t.

 


 

 


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The Lighter Side

 

NEWS

There’s something about Gödel by Francesco Berto

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.

 

 

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.

 

 

Updated: Diagonal Lemma

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

Previous Blog Posts  

 

16th Mar 2015 Bishops Dancing with Pixies?

 

23rd Feb 2015 Artificial Intelligence

 

31 Mar 2015 Cranks and Crackpots

 

30 Apr 2015 The Chinese Room

 

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

 

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