Oh no ! Yet Another Flawed Incompleteness Proof
From the collection of obviously flawed incompleteness proofs, here is yet another:
A Flawed Incompleteness Proof by Bernd Buldt
Dr. Bernd W. Buldt is Professor of Philosophy at Purdue university, where he teaches “Gödel’s Incompleteness Theorem”, as well as other topics. One of Buldt’s publications is an article on Gödel’s first incompleteness theorem, written for a summer school, online at PDF UniLog5 - Tutorial 1 - Bernd Buldt.
Buldt also has similar articles online at PDF Gödel’s First Incompleteness Theorem: Tutorial 2 - Bernd Buldt, and PDF Gödel’s First Incompleteness Theorem: Tutorial 3 - Bernd Buldt). There is also a review of Buldt’s “The Scope of Gödel’s First Incompleteness Theorem” (Footnote: Buldt, Bernd, The Scope of Gödel’s First Incompleteness Theorem, Logica Universalis 8.3-4 (2014): pp. 499-552. Details at The Scope of Gödel’s First Incompleteness Theorem.) at A review of Buldt’s The Scope of Gödel’s First Incompleteness Theorem.
Other obviously flawed incompleteness proofs can be seen at:
Buldt states that his areas of interest include Mathematical Logic and Philosophy of Mathematics, so we might expect his work to be logically rigorous. Unfortunately his Tutorial 1 on “Gödel’s Incompleteness Theorem” demonstrates the complete opposite, as will be shown below.
gn(v) is a term of the meta-language which gives the Gödel number, in the format of the meta-language, of the formal formula v.
n is a term of the meta-language which converts a number n that is in the format of the meta-language into the format for numbers of the formal system. (Footnote: The initial n should appear with a bar over it. If no bar is showing, your browser is not displaying the content as intended. Perhaps you have the CSS styling turned off in your browser.)
⌈φ⌉ is an expression of the meta-language which gives the Gödel number, in the format of the formal system, of the formal formula φ. Note that ⌈φ⌉ = gn(φ).
In the section Fixed point derivation, Step 1: Substitution Buldt says that, for a given variable u of the formal system, the aim is to:
Define a function sub that mirrors the substitution of the replacee variable ‘u’ for a replacer term ‘t’.
Unfortunately, his “definition” of his function sub(x, y) is rather confused (of which more later). He states that if x is the Gödel number of a formal system formula φ(u) with one free variable u and if y is the Gödel number of t where t is a term of the formal system, then the value of sub(x, y) is the Gödel number of the formal system formula that results from the substitution of the free variable u of φ(u) by t. Otherwise the value of sub(x, y) is simply x.
Buldt claims that sub(x, y) is primitive recursive and that there is a corresponding expression in the formal system. He provides no proof whatsoever of this crucial claim. But the fact is that the definition that Buldt has provided is completely dependent on Gödel numbering, which is a function that is not primitive recursive, since it is not number-theoretic, and so, unless he had provided an alternative definition of his sub(x, y) - which he didn’t - then the sub(x, y) that he did define is not primitive recursive. I see this time and time again in so-called proofs of incompleteness, where a key statement is conveniently assumed when it looks as if actually providing a proof might be impossible.
In the section Fixed point derivation, Step 2: Definitions, Buldt defines:
φ(u) :≡ ∀x[¬ProofF(x, sub(u, u))]
Now, let’s try to see how we can apply the definition of sub(x, y) to sub(u, u). We get that if u is the Gödel number of a formal system formula φ(v) with one free variable v and if u is the Gödel number of t where t is a term of the formal system, then the value of sub(u, u) is the Gödel number of the formal system formula that results from the substitution of the free variable v of φ(v) by t. Otherwise the value of sub(u, u) is simply u.
Well, clearly there is a problem here. The function p is only defined for p as a number, but of course, if u is the Gödel number of a formal system formula φ(v) then u cannot at the same time be the Gödel number of a number. But maybe it will all become clear as we work through Buldt’s text? Let’s see. Buldt now tells us that when we substitute p for u in φ(u) we get:
γ :≡ φ( p ) ≡ ∀x[¬ProofF(x, sub( p, p ))]
Further on in the same section, we have that sub(p, p) = gn(γ) and if sub( p, p ) is the equivalent expression in the formal system, then we would have:
(i) sub( p, p ) = gn(γ) = ⌈γ⌉
(ii) ⌈γ⌉ = ⌈ ∀x[¬ProofF(x, sub( p, p ))] ⌉
and from (i) we substitute ⌈γ⌉ for sub( p, p ) in (ii) above, this gives:
⌈γ⌉ = ⌈ ∀x[¬ProofF ( x, ⌈γ⌉ )] ⌉
which is impossible, since, on the one hand, the numerical value on the right-hand side must be greater than that of the left-hand side, since the right-hand side includes the numerical value ⌈γ⌉ of the left-hand side within it, but on the other hand, by the equality, the numerical value of the right-hand side must be equal to the numerical value of the left-hand side.
Buldt attempts to get around this problem in the next section Fixed point derivation: Redux by asserting that some expressions that are defined in terms of the Gödel numbering function can be represented in some other way in the formal system by a relation although there is not an exact correspondence as was hitherto supposed. He calls this relation sub′( p, p, ⌈γ⌉), and he declines give any definition of this relation, which is very convenient for Buldt, since by declining to do so he evades the crucial question of how the formal system can have information about a Gödel numbering coding system that operates completely outside the system.
In fact, if you actually analyze what Buldt does say about this relation that he assumes can exist within the formal system, we see that since:
p = gn(φ(u))
p = gn(φ(u)) = ⌈φ(u)⌉
γ = φ( p ) = φ(⌈φ(u)⌉)
⌈γ⌉ = ⌈φ(⌈φ(u)⌉)⌉
we find that the relation is:
sub′(⌈φ(u)⌉, ⌈φ(u)⌉, ⌈φ(⌈φ(u)⌉)⌉)
and we can easily see that Buldt is claiming that there is a relation in the formal system that asserts a relation between the Gödel number of a formula φ(u) of the formal system, and the Gödel number of that same formula φ(u) when the free variable u of that formula has been substituted by the Gödel number ⌈φ(u)⌉ of that formula φ(u).
Yes - it’s a lot easier to assume that there is such a relation in the formal system than to actually prove that there is one - especially if it is impossible to prove. (Footnote: See the paper PDF The Impossibility of Representation of a Gödel Numbering Function by a Formula of the Formal System.) Simply assume that the formal system can self-reference, rather than actually prove that it can, and of course you can end up with all sorts of bizarre results, as Buldt does. It’s also bizarre that people actually pay to be subjected to this sort of hocus-pocus.
There’s nothing profound nor logical about Buldt’s combination of simple-minded assumptions and lines of equations that appear impressive to the uninitiated. And despite the patently obvious assumption of self-reference in the above, Buldt’s risible failure to prove the crucial points of his “proof ” leads him to the inane conclusion:
“There is no self-reference at work but only simple or multiple diagonalization; Fixed points are equivalent, not self-referential in any strong sense we could make precise; we are not skating on the thin ice of paradox.”
No, not skating on the thin ice of paradox - only wallowing in self-delusion.
Also see Errors in incompleteness proofs and Analysis of incompleteness proofs.
Other obviously flawed incompleteness proofs can be seen at:
An Incompleteness Proof by Francesco Berto
An Incompleteness Proof by Dan Gusfield
An Incompleteness Proof by Byunghan Kim
An Incompleteness Proof by Dennis Müller
An Incompleteness Proof by Arindama Singh
An Incompleteness Proof by Sebastian Oberhoff
An Incompleteness Proof by Antti Valmari
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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