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Copyright   James R Meyer    2012 - 2024 https://www.jamesrmeyer.com

Oh no ! Yet Another Flawed Incompleteness Proof

Page last updated 23 Jun 2021


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

A Flawed Incompleteness Proof by Dennis Müller

Dennis Müller, a student in computer science, is so enamored with the notion of an incompleteness theorem that he has a statement of it tattooed on his arm, see Why I tattooed Gödel on my arm. Apparently he has done so because he believes that it has “profound philosophical implications”.


Dennis Muller's incompleteness tattoo
No - it is rather like having “You cannot prove this statement” tattooed on your arm. Rather than being profound, both are examples of rather childish word play based on confusion of language.


That anyone would think that a statement stating something similar is stating something profound perhaps tells you more about them than it does about real mathematics.


If you want a profound mathematical statement there are plenty available - for example, the brief but timelessly profound equation Euler’s identity:

ei π =  −1

is, unlike any incompleteness statement, the underlying foundation of much of science and technology.


Müller has written a brief “proof ” of incompleteness that results in the statement tattooed on his arm. He has written a disclaimer of his “proof ”, warning that it is “just a simplification”. While I can accept that an author might leave out details and notify the reader that one has to accept the claim that certain statements have been proved elsewhere, but regardless of whether it is a simplification, every statement that he makes must be logically valid. Otherwise what’s the point? - you might as well say, “here’s an illogical and erroneous proof that I’m showing you in the expectation that you will then believe that somewhere there is a valid error-free proof.”


Anyway, Müller proceeds to explain how Gödel numbering can be used to create a correspondence between sequences of a formal system and Gödel numbers. Then he introduces a purely number-theoretic function called sub(n, m), of which he says that while n is a Gödel number of a formal system formula F(x)m is not necessarily a Gödel number, and yet sub(n, m) is supposedly the Gödel number of the formula that results when the free variable of the formula F(x) whose Gödel number is n, is substituted, not as one would expect, by the sequence whose Gödel number is m, but by the actual number m.


In doing so, Müller has just thrown the entire raisin d’etre of Gödel numbering out of the window - which is that Gödel numbering should provide a precise correspondence between: (Footnote: One sees similar fudges regarding a substitution function turning up time and time again in various attempts at incompleteness proofs, see for example Gödel’s Substitution Function, Hofstadter’s ‘Gödel, Escher, Bach’, Nagel & Newman’s Book: Gödel’s Proof.)

  1. relationships between formal sequences and
  2. relationships between the corresponding Gödel numbers.


Later on, we see that what Müller actually meant was that his sub function should correspond to the substitution of the free variable of a formula F(x) by the Gödel number of that formula F(x). If that is so, then one can have a purely number-theoretic function with two free variables, where by substitution, one corresponds by Gödel numbering to the formula F(x), and the other corresponds by Gödel numbering to the Gödel number of the Gödel number of F(x).


But Müller assumes that a single variable can replace the two variables n and m, and in doing so, he is assuming that the purely number-theoretic expression can express the notion of Gödel numbering. It can’t, since Gödel numbering is not a purely number-theoretic function. Müller has completely sidestepped this problem, and only the simple-minded will be taken in by this fudge.


Müller finishes up with a eulogy, claiming that incompleteness proofs represent “mathematics at its absolute best ” and that “the incompleteness theorems are to logic (and logic is to math), as Heisenberg’s uncertainty principle is to quantum mechanics (and qm is to physics).”


No - it is pseudo-mathematics at its absolute worst - and attempts at incompleteness theorems are to logic as New Age mysticism is to logic - i.e: they are the antithesis of logic.


In summary, although Müller includes a disclaimer, the fact is that his ‘explanation’ doesn’t actually explain anything, instead it raises more questions than it answers.


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