Logic and Language

Logic and Language

Copyright © James R Meyer 2012 - 2018 https://www.jamesrmeyer.com

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From the collection of obviously flawed incompleteness proofs, here is yet another:

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 Gödel’s First Incompleteness Theorem: Tutorial 1 - Bernd Buldt, and also at Fixed Points, Diagonalization, Self-Reference, Paradox and UniLog5 - Tutorial 1 - Bernd Buldt.

Buldt also has similar articles online at Gödel’s First Incompleteness Theorem: Tutorial 2 - Bernd Buldt, and 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. Online 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:*

An Incompleteness Proof by Francesco Berto

An Incompleteness Proof by Dan Gusfield

An Incompleteness Proof by Byunghan Kim

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.

Preliminary notes:

is a term of the meta-language which gives the Gödel number, in the format of the meta-language, of the formal formula*gn*(v)**v**.-
**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.) **⌈**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***⌈**.*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

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

In the section *Fixed point derivation, Step 2: Definitions*, Buldt defines:

*φ*(u) :≡ ∀x[¬Proof_{F}(x, sub(u, u))]

Now, let’s try to see how we can apply the definition of ** sub(x, y)** to

Well, clearly there is a problem here. The function ** is only defined for ****t** as a number, but of course, if ** u** is the Gödel number of a formal system formula

**γ :≡ φ( ) ≡ ∀x[¬Proof_{F}(x, sub( , ))]**

Further on in the same section, we have that ** sub(p, p) = gn(γ)** and if

(i) **sub( ** , ) = = ⌈γ⌉

and so

(ii) **⌈γ⌉ = ⌈ ∀x[¬Proof _{F}(x, sub( , ))] ⌉**

and from (i) we substitute **⌈γ⌉** for **sub( ** in (ii) above, this gives: , )

**⌈γ⌉ = ⌈ ∀x[¬Proof _{F} ( 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′( **, 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. , , ⌈γ⌉)

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))**

** φ(u)⌉** = = ⌈

**γ = φ( ) = 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

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 The Impossibility of Representation of a Gödel Numbering Function by a Formula of the Formal System (PDF).) 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 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.

Footnotes:

*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

Diverse opinions and criticisms are welcome, but messages that are frivolous, irrelevant or devoid of logical basis will be blocked. Difficulties in understanding the site content are usually best addressed by contacting me by e-mail. Note: you will be asked to provide an e-mail address - any address will do, it does not require verification. Your e-mail will only be used to notify you of replies to your comments - it will never be used for any other purpose and will not be displayed. If you cannot see any comments below, see Why isn’t the comment box loading?.

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

https://www.jamesrmeyer.com