Copyright © James R Meyer 2012 - 2018 https://www.jamesrmeyer.com
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(x) = GN(x)’
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 GN(x) is not an expression in the language of number relationships and not an expression in the formal language – it is an expression of the meta-language. And that means that the variable x in GN(x) is a variable of the meta-language.
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 (GS, BN(GS))’
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(x) = GN(x)’ in order for his proof to work.
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(x) = GN(x)’
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(x) = GN(x)’
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.
See also Gödel’s contradiction.
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There is now a paper that deals with the matter of language and the diagonal proof, see On Considerations of Language in the Diagonal Proof.
There is now a new page on a contradiction in Lebesgue measure theory.
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.
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