Logic and Language

Logic and Language

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

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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

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,

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

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}**.

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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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