This page is keyboard accessible:
• Use Tab, Shift + Tab keys to traverse the main menu. To enter a sub-menu use the Right Arrow key. To leave a sub-menu use the Left Arrow or the Escape key.
• The Enter or the Space key opens the active menu item.
• To skip the menu and move to the main content, press Tab after the page loads to reveal a skip button.
• To get back to the top of the page anytime, press the Home key.
• For more information, click here: Accessibility   Close this tip.

Note: Full functionality of this web page requires JavaScript to be enabled in your browser.

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 The Scope of Gödel’s First Incompleteness Theorem)


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.



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.

section divider


section divider



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?.

section divider

The Lighter Side


Paper on the diagonal proof

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.

section divider

Other recently added pages

The Myths of Platonism


Goodman’s Paradox


The Platonist Rod paradox


The Balls in the Urn Paradox


section divider

Lebesgue Measure

There is now a new page on a contradiction in Lebesgue measure theory.

section divider

Easy Footnotes

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).

section divider

O’Connor’s “computer checked” proof

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.

Previous Blog Posts

Moderate Platonism

Descartes’ Platonism

The duplicity of Mark Chu-Carroll

A John Searle Inanity

Man versus Machine

Fake News and Fake Mathematics

Ned Block’s Blockhead

Are we alone in the Universe?

Good Math, Bad Math?

Bishops Dancing with Pixies?

Artificial Intelligence

Cranks and Crackpots

The Chinese Room


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:

Gödel Links


– and a page relating specifically to the Gödel mind-machine debate:

Gödel, Minds, and Machines

Printer Friendly

All pages on this website are printer friendly, and will print the main content in a convenient format. Note that the margins are set by your browser print settings.

Note: for some browsers JavaScript must be enabled for this to operate correctly.


Comments on this site are welcome, please see the comment section.


Please note that this web site, like any other is a collection of various statements. Not all of this web site is intended to be factual. Some of it is personal opinion or interpretation.


If you prefer to ask me directly about the material on this site, please send me an e-mail with your query, and I will attempt to reply promptly.


Feedback about site design would also be appreciated so that I can improve the site.

Copyright © James R Meyer 2012 - 2018