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A review of Buldt’s The Scope of Gödel’s First Incompleteness Theorem

The following is a review of a draft version of Bernd Buldt’s “The Scope of Gödel’s First Incompleteness Theorem”. The review was requested by the publishers of Logica Universalis. Despite my review indicating many deficiencies in the draft, the paper was published later in that journal without those deficiencies being corrected. (Footnote: Buldt, Bernd, The Scope of Gödel’s First Incompleteness Theorem, Logica Universalis 8.3-4 (2014): pp. 499-552. Online at https://opus.ipfw.edu/philos_facpubs/298/)

 

Review of the draft paper

The author makes significant errors. An early error (section 1.6) is an incorrect assertion regarding primitive recursive functions. The author provides an initial exposition of a combination of Gödel’s original 1931 proof and some notes taken of Gödel’s lectures (1934). The author then introduces a function “sub” and makes the false assertion that sub(x,y) is primitive recursive. Primitive recursive functions are functions that are necessarily number-theoretic functions, that is, they are functions whose objects are natural numbers, and whose variables (free and bound) have the domain only of natural numbers. But the author’s sub function is defined in terms of variables whose domains include entities other than natural numbers, and thus the sub function is not defined in number-theoretic terms and is thus not primitive recursive (that is, in order to evaluate the function requires the use of variables whose domain is not natural numbers).

 

The author may be making the common mistake of assuming that if a function’s free variables have the domain of natural numbers, and if the function has the range of natural numbers, then it is a number-theoretic function. This is not the case. Furthermore, since the proof relies on the proposition that for every primitive recursive number-theoretic relation/function, there is a corresponding formula of the formal system that expresses that function, it is obvious that the entire definition of a relation/function must be in purely number-theoretic terms in order that this can be the case (and which includes all bound variables as number-theoretic variables).

 

It might argued that it might be possible that the sub function can be defined in purely number-theoretic terms, but the author has failed to do so. Since the remainder of his proof relies on the function sub being a primitive recursive number-theoretic function, the proof is not logically valid. This error appears to indicate fundamental deficiencies in the authors understanding. Furthermore, since most of the paper relies on this proof, such an error is not of little consequence.

 

Another error is the claim that Gödel’s proof of the first incompleteness theorem is intuitionistically valid - The Corollary 1.3 (incidentally Corollary is spelt incorrectly). The author appears to base this claim on the fact that a particular formula of the formal system can be constructed from the given proof. It appears that the author simply assumes that constructibility of a specific entity mentioned in a proof implies that the entire proof can be proved using intuitionistic logic. But Gödel’s proof relies on, among other things, a sketch of a long involved proof, and the author provides no logical argument to demonstrate that the fully completed proof would necessarily satisfy intuitionistic logic.

 

The author argues self-reference is not necessarily involved in incompleteness proofs (section 2.2, pp 17-18). The author himself states further on, “There is not a single proof for G1 known (to this author) that would not rely on diagonalization”; and although the author does not define precisely what he means by the term diagonalization, his references to it indicate that it is inherently self-referential. The author claims that his “double diagonalization” does not involve “proper” self-reference, but clearly his “double diagonalization” does involve self-reference, since it involves a formula being substituted by an entity which, for certain substitutions will be an entity that directly refers to the formula itself that is being substituted.

 

The author refers to “proper” and “true” self-reference, but he provides no elucidation on what he intends by such terms. Perhaps the author intends that a “true” or “proper” self-reference is one that is explicit in an expression at the end of the proof process. But that does not alter the essential fact of self-reference being introduced at some point in the proof process, and whether the self-reference appears explicitly in certain expressions is simply a matter of symbolic manipulation.

 

The author refers to papers claiming that they support his contention regarding self-reference (Halbach/Visser and Adamowicz/Bigorajska). But Halbach/Visser’s arguments relate to direct explicit self-reference, so they do not support his general contention regarding self-reference - in fact Halbach/Visser actually state, “in this paper self-reference is always understood [as] direct self-reference”. The Adamowicz/Bigorajska argument is less than convincing - it argues that self reference is a result of both diagonalization and substitution together, and then, given that proposition, it follows that diagonalization on its own is not self-referential - which is simply a case of picking the antecedent to give the desired result.

 

The author also refers to [Lawvere 1969], [Yanofsky 2003], [Bell 2007] to support his contentions regarding self-reference, but there is nothing in those papers to support his claims. A well-reasoned argument that stands on its own, with or without a few pertinent references, is always preferable to vague hand-waving with irrelevant references that waste the time of a reviewer.

 

Rating:

The following subjects should be ranked from 1 to 10 (High 9-10, Good 7-8, Average 5-6, Poor 3-4, Low 0-2), following by some comments.

 

Originality and novelty: 5 - Much of what appears in the paper has been expressed elsewhere.

 

Importance: 4 - The errors in the paper render it virtually worthless

 

Technical soundness: 4 - Too many technical errors that are not simply oversights

 

Quality of writing and style (presentation): 7 - Generally fair, although the text is frequently irritating to read, because the author frequently refers to terms which he has not previously defined or explained. For example, in Section 1.7, he writes “Recall the idea behind omega-consistency ...”, and yet this is the first mention of omega-consistency in the paper.

 

Relevance for this special issue of the journal:

Relevance not applicable, since the significant errors in the paper render it unsuitable for publication.

 

Overall judgment: 3 - Since most of the paper relies on a flawed proof, I do not consider it acceptable for publication.

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Footnotes:

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A review of Buldt’s The Scope of Gödel’s First Incompleteness Theorem

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O’Connor’s “computer checked” proof

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