Uwe Petersen on Gödel’s incompleteness proof

Page last updated 16 Sept 2026

I give here an analysis of a paper by Uwe Petersen, ‘On an Alleged “Fatal Flaw” in Gödel’s Incompleteness Proof,’ that he published Nov 2025 on that most disreputable site Acaedmia.edu, whose score on Trustpilot is 1.1. Part of the 0.1 of the score is due to reviewers who have given 5 stars to every company they review. You have been warned: Do not pay anything to that site.

Note that in the following, Petersen’s remarks are in green text. Petersen’s paper is a rambling discourse with little in the way of critical logical analysis, much of it relying on Petersen’s creation of straw men that he then proceeds to demolish. The irony is that at the outset he includes the quote:

“Rubbish is rubbish, but the history of rubbish is scholarly.”

Petersen wastes no time in adding to the mountain of rubbish that has a scholarly veneer.

In the abstract he says:

The point in a nutshell: the formation of the function Sb(a₁, a₂, Z(a₃ ))) is illegitimate according to Meyer, but [it is] in perfect agreement with the formation rules for primitive recursive functions, which Meyer fails to take into account.

That is factually wrong; I never claimed that the formation of that function is illegitimate, it is a perfectly valid primitive recursive function. What I do claim is that there is an illogical and unsustainable assumption of an equivalence between that function and a function Sb(a₁, a₂, a₄ ) - a function which corresponds, by Gödel numbering, to the substitution of the free variable of the formal expression A₁ by the formal expression A₄ , where a₁ is the Gödel number of A₁ and a₄ is the Gödel number of A₄ .

Petersen then says:

With regard to 2. above, what Gödel actually does is to create a correspondence between predicates regarding formal expressions and number-theoretic predicates, where this correspondence is generated by the Gödel numbering function (as referred to in 1. above). The number-theoretic predicates of themselves do not express the predicates regarding formal expressions, it is only the combination of those number-theoretic predicates and Gödel numbering that can give the information regarding the predicates regarding formal expressions.

Petersen goes on to state the obvious, that a primitive recursive function can be defined in terms of multiple prior primitive recursive functions. Then he quotes from my webpage Gödel’s Substitution Function:

Petersen then adds his commentary:

Now that’s the point where Meyer goes badly wrong. It is not just because of the claim that “the value given for third variable y of the function must be declared, outside of the formal system” which has no basis in Gödel’s paper, and the phrasing “the free variables of Sb must be Gödel numbers” which may at best mean that the free variables of Sb can only take numbers as values.

Oh dear, oh dear. Presumably Gödel assumed that any competent reader would understand how he uses his numbering function, but Petersen displays a elementary misunderstanding. As noted above, Gödel’s numbering function operates to create a correspondence between, on the one hand, relationships between formal expressions and, on the other hand, relationships between numbers where those number are the corresponding Gödel numbers of those formal expressions. For if it were the case that in any given Sb(x, v, y) where the x, v and y are specific numbers, that x and y are not Gödel numbers then Sb(x, v, y) cannot correspond by Gödel’s system to the substitution of the free variable of any formal expression by some formal expression.

And here we touch on a fact regarding Gödel numbering which is almost universally overlooked - one can have a number-theoretic expression which corresponds by Gödel numbering to a relation between expressions of the formal system, but which also contains a number that simply happens to have the format of a Gödel number, but which has no corresponding formal expression. This means that one must maintain a very precise track of the correspondences given by every introduction of a number-theoretic relation that involves Gödel numbering - otherwise one can have a spurious appearance of a correspondence to a relationship between formal expressions simply by the fact that a number happens to have the format of a Gödel number. And that is precisely what happens when the Z function is used within the Sb function.

Incidentally, Petersen complains:

There is no such term as GN(Y) in Gödel’s paper.

But I never claimed there was. I used the term GN( ) on that webpage simply for convenience to indicate the Gödel numbering function, and I clearly state that on that page when I introduce the term. Petersen’s complaint is either an indication of laziness of investigation or else willful duplicity.

Petersen goes on to quote from my Formal Paper: The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem:

Petersen remarks that: “…I didn’t find any such “definition” relating to italicized words.”

In fact, Gödel’s original German paper specifically refers to using italicized words for Gödel numbers to indicate the type of formal expression that the number corresponds to by such numbering, and throughout his original German paper, given the term for a formal expression such as “formula”, he uses the italicized form of the same term (i.e: formula) to refer to the Gödel number that corresponds to that formal expression. So it’s another red herring from Petersen. Note that in Van Heijenoort changes the italicization to words in small capital letters in his translation, which appears in the book: “From Frege to Gödel: A Source Book in Mathematical Logic”, publisher: Harvard University Press.

Petersen then says:

Meyer’s strategy just strikes me as an attempt to distract from Gödel’s painstaking definitions of meta-theoretical predicates as primitive recursive functions and relations, so he can shift considerations from a “precise language, without reference to any interpretation of the formulas” to an intuitive interpretation of the formulas in order to be able to discover a confusion of language levels.

It’s very convenient for Petersen to make such claims without actually providing any instances at all of what he is claiming. Where in my formal paper is there any such “intuitive interpretation”? The fact is that in my formal paper, I make a critical logical analysis and I do not rely in any way on interpretations, and Petersen’s claim is downright reprehensibly and despicably dishonest. Furthermore, although Petersen appears fixated on primitive recursion, it has absolutely nothing to do with the creation of a precise correspondence (as noted above) between predicates regarding formal expressions and number-theoretic predicates where this correspondence is generated by the Gödel numbering function. It is this precise correspondence, not primitive recursion, that enables a purely logical non-intuitive analysis of the concepts given by the combination of various number-theoretic expressions and Gödel numbering. We can also note that Gödel refers extensively to his Bew relation in his proof, a relation which, as he remarks, cannot be asserted to be primitive recursive.

Petersen then makes an obvious error:

One last example of Meyer’s obsession with the equivalence of two functions one of which does not even occur in Gödel’s proof:

and then quotes again from my Formal Paper: The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem:

The assertion gives rise to the question of how such an equivalence can apply if the formats of the numbers that constitute the domains of the free variables of Z function and his numbering function Φ(n) are different. However, we can consider for a moment the case where the format for numbers of the domain of the free variable of the Z function is precisely identical to that of the formal system P of Gödel’s paper, which gives the assertion of equivalence as applying for the domain of the free variable n as natural numbers in that format: Z(n) = Φ(n).

But the blatant fact is that the other function, the Φ function does appear explicitly in Gödel’s original paper and in translations. And even if Gödel had not explicitly named that function, the obvious fact is that he defines that function in his paper. It seems to be Petersen who is blinded by some sort of obsessional need to try to rescue Gödel’s paper. He goes on to say:

Contrary to Meyer’s claim, Gödel’s proof doesn’t involve the Φ-function which is indeed not a function that can be formulated within the arithmetic system.

This is arrant nonsense - if Gödel’s proof doesn’t involve the Φ-function then all references to it in his paper could be removed without affecting the proof ! - that isn’t logic, it is complete and utter bullshit.

Petersen goes on to complain that Gödel does not explicitly state an equality between the Z function and his numbering function (and again makes further irrelevant quibbles about my use of the term GN to indicate the Gödel numbering function). But I provide ample proof elsewhere that Gödel did indeed intend such an equivalence, and indeed the proof necessarily requires that assumption of equivalence, see The Flaw in Gödel’s Proof of his Incompleteness Theorem, so I will not repeat that content here.

Petersen continues to bang on ad nauseam about primitive recursion as if nothing else is involved in Gödel’s proof, again quoting from my Formal Paper: The Fundamental Flaw in Gödel’s Proof of his Incompleteness Theorem:

…the definition of the terms ‘recursive’ and ‘ω-consistent’ are not here defined, since these terms are defined in Gödel’s paper, and their precise definition is immaterial to the argument here presented.

and then he declares:

That says it all. While the concept of primitive recursive function is of crucial importance in Gödel’s proof, it is immaterial to Meyer’s argument.

Petersen thereby indicates a complete failure to understand that an argument that does not rely on the precise definition of a particular matter does not imply that there is anything wrong with that argument. It merely means that the argument stands on its own without having to rely on the details of any such precise definition.

Finally Petersen digs a hole for himself when he suggests that confusion is something to be applauded in mathematics:

What Gödel has achieved is an immaculate confusion of use and mention which is based on a clever application of primitive recursive functions.

and also:

So let me emphasize (again?), it is the “precise definition” of the primitive recursive functions, specifically the justification of Z occupying an argument place in Sb, which enables Gödel’s impeccable confusion of use and mention.

Well, given an expression that is constructed from the alphabet of a given language, “use” is when that expression is used as part of the syntax of the language, while “mention” is when that expression is referred to, but which does not have those syntactical properties as when it is used in syntax. Hence a confusion of use and mention is to confuse an expression which has certain properties with an expression which does not have those properties. Petersen fails to see that that might be problematic when one is trying to create a precise logical argument. One can only despair when one sees such idiocy masquerading as logic.