Oh no ! Yet Another Flawed Incompleteness Proof

Francesco Berto is a professor of philosophy at the University of St Andrews and is a member of the Institute for Logic, Language and Computation (ILLC). He has written a book that he claims is “The Complete Guide to the Incompleteness Theorem”. (Footnote: Francesco Berto: There’s Something about Gödel: The Complete Guide to the Incompleteness Theorem, ISBN: 978-1-4051-9766-3, Wiley-Blackwell, 2009 There’s Something about Gödel: Details.) When I was asked to if I could see anything wrong with Berto’s argument, I took a quick look at the book and within a few minutes I found the error in it. It was easy because Berto essentially just rehashes the content of Nagel & Newman’s book, (Footnote: PDF E Nagel and J Newman: Gödel’s Proof New York University Press, revised edition, 2001. ISBN: 0814758169.) without actually understanding it. (Footnote: The content of Nagel & Newman’s book is analyzed on another webpage Nagel & Newman’s Book: Gödel’s Proof.)

As such Berto’s book is not by any stretch of the imagination any sort of reliable guide to Gödel’s Incompleteness proof, never mind a ‘complete’ guide. For Berto to call his book a complete guide when it is boils down to a rehash of a 50 year old informal exposition of Gödel’s proof is, quite frankly, ridiculous. But, worse than that, Berto demonstrates that he does not even understand the subtleties involved in Nagel & Newman’s account.

 

In Section 3, “Arithmetizing substitution” in Chapter 6, “I Am Not Provable”, Berto defines a number-theoretic function as follows:

Let us now consider an arithmetic function, Sub(m, n, p), such that, if m is the Gödel number of a TNT formula α, and n is the Gödel number of a variable x, its value is the Gödel number of the formula α[x/p]. This is the formula one obtains from α (whose code is m), by replacing in it the free occurrences of the variable x (whose code is n) with the term p, that is, with the numeral of number p.

 

Berto then goes on to say:

We can read “Sub(m, n, p)” as: “the code of the formula obtained from the formula whose code is m, by substituting in it the free occurrences of the variable whose code is n with the numeral of p. And one can show that Sub is primitive recursive”

 

But in the very next paragraph, Berto gives a quite different definition of that very same function, referring to:

v = Sub(t, 31, t)

and stating that v is the Gödel number that corresponds to the formula one obtains from the formula whose Gödel number is t, when the free variable in that formula (which corresponds by Gödel coding to 31) is substituted by “the numeral of the Gödel number of t itself ”.

 

Note the discrepancy here - in the first definition, the definition that asserts that the function Sub is purely number-theoretic, the corresponding substitution is simply by the numeral of a given number. But in the second definition, the corresponding substitution is by the numeral of the Gödel number of a given number. And it is the second definition that Berto uses to “prove” incompleteness, by assuming that the formal system TNT can express the second definition. But the second definition is not a purely number-theoretic expression and it can be proved that there cannot be an expression in the formal system TNT that can express the second definition. (Footnote: See PDF The Impossibility of Representation of a Gödel Numbering Function by a Formula of the Formal System.)

 

Clearly, Berto has simply copied snippets of the content of Nagel & Newman’s book without subjecting it to any sort of critical logical analysis, and without understanding precisely what he is doing. He does not seem to be aware that he has used two different definitions of the same function. It has to be said that Nagel & Newman’s description of their Sub function is vague and verbose and in places contradictory, and one has to read it carefully to ascertain that Nagel & Newman are including the implicit assumption that the formal system can refer to the Gödel coding function - which it cannot. (Footnote: See PDF The Impossibility of Representation of a Gödel Numbering Function by a Formula of the Formal System.) But if Berto is asserting that he has produced a complete guide to Gödel, then it was incumbent of him to critically analyze what he was writing. Berto’s error is so obvious that it would seem that Berto is incompetent or careless or dishonest - or some combination of these attributes.

 

What is also surprising is that none of the reviews that Berto’s book that I have seen makes any mention of the obvious error of two different definitions of the same function. (Footnote: ▪ A review by The Times Higher Education, see Review of There’s Something About Gödel. ▪ A review by Vann McGee, see Review of Francesco Berto’s There’s Something about Gödel. McGee has written a ‘proof’ of the Diagonal Lemma; the analysis of his ‘proof’ is at Vann McGee’s Proof of the Diagonal Lemma. ▪ A review by the convicted paedophile ex-professor Peter Smith, see Review of Berto: There’s Something About Gödel. Smith has published a book on Gödel’s proof, which contains a fatal flaw; the analysis of the Smith’s book is given at PDF A Fundamental Flaw in an Incompleteness Proof by Peter Smith. Smith also has a downloadable cut-down version of his book which is dealt with on the page Peter Smith’s Gödel Without (Too Many) Tears - Or Not?.) Does the fact that these reviewers were all unable to see this elementary error indicate that they are dim-witted or that they are careless? Or is it just a case of confirmation bias, where the reviewers are not impartial, and simply saw what they wanted to see, rather than subject Berto’s content to logical scrutiny?

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