A Simplified Explanation of Gödel’s proof - Part 10
10: A Demonstration of a Contradiction Arising from Gödel’s Proof
Page last updated 18 Feb 2022
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.
You can see an in-depth examination of this assumption of equivalence at The Flaw in Gödel’s proof of his Incompleteness theorem; note that the function represented on this page by BN(x) is referred to on that page as Z(n), while the function represented on this page by GN(x) is referred to on that page as Φ(n).
See also the page Gödel’s Proposition V, which shows that Gödel’s Proposition V leads to a blatant contradiction, and the page Gödel’s contradiction. And for a fully detailed formal analysis of Gödel’s paper, see PDF The Fundamental flaw in Gödel’s proof, where there is now an additional section that gives a brief summary of the underlying illogical assumption that the proof relies on, so that the reader can see in a few pages that the proof is flawed.
Rationale: Every logical argument must be defined in some language, and every language has limitations. Attempting to construct a logical argument while ignoring how the limitations of language might affect that argument is a bizarre approach. The correct acknowledgment of the interactions of logic and language explains almost all of the paradoxes, and resolves almost all of the contradictions, conundrums, and contentious issues in modern philosophy and mathematics.
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