Logic and Language

Logic and Language

Copyright © James R Meyer 2012 - 2016 www.jamesrmeyer.com

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From time to time I get asked to comment on an incompleteness proof. Many of these proofs are so obviously flawed that it is easy to demonstrate where the flaw is. It is not worth having a separate web page for each such proof, so I decided to start this page to deal with proofs of incompleteness that are so obviously flawed that the flaw can be explained in a few paragraphs, and if I have time, I will populate this page with suitable examples. It seems to be the case that in many such proofs, the quality of the proof seems to be in inverse proportion to the esteem with which the authors view their own work.

The proofs so far dealt with are:

- An Incompleteness Proof by Francesco Berto
- An Incompleteness Proof by Arindama Singh
- An Incompleteness Proof by Antti Valmari
- An Incompleteness Proof by Dan Gusfield
- An Incompleteness Proof by Byunghan Kim

Francesco Berto is a professor of philosophy at the University of Amsterdam 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 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: E Nagel and J Newman. *Gödel’s Proof*. New York University Press, revised edition, 2001. ISBN: 0814758169 Gödel’s Proof: Details.) without actually understanding it. (Footnote: The content of Nagel & Newman’s book is analyzed on another web-page 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 Theorem, 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 The Impossibility of Representation of a Gödel Numbering Function by a Formula of the Formal System (PDF).)

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 The Impossibility of Representation of a Gödel Numbering Function by a Formula of the Formal System (PDF).) 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 here.) (Footnote: A review by Vann McGee, see here. McGee has written a ‘proof’ of the Diagonal Lemma; the analysis of his ‘proof’ is at Vann McGee’s Proof of the Diagonal Lemma.) (Footnote: A review by Peter Smith, see here. Smith has published a book on Gödel’s proof, which contains a fatal flaw; the analysis of the Smith’s book is given at A Fundamental Flaw in an Incompleteness Proof by Peter Smith PDF.) 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?

Footnotes:

Arindama Singh is a professor in the Department of Mathematics in the Indian Institute of Technology Madras and has written *Foundations of Logic* and *Logics for Computer Science*. He has also written a paper A Simple Proof of Gödel’s Incompleteness Theorem which includes several elementary logical errors - this paper was published in the Ramanujan Mathematical Society’s Mathematics Newsletter Volume 22. No3, December 2012.

Early on in the paper, Singh refers to the standard turnstile symbol and acknowledges its standard use when he states, regarding a formal system of arithmetic * N*, that we write:

* X*

to signify “** X** is a theorem in

He then says that **P****( X) = Pr(g(X))**, where

Now, ** Pr(g(X))** is a meta-language expression that is intended to

i.e., if ** x = g(X)**, then

“* X *is provable in

and also to the assertion **P****( X)**.

Singh’s assertion that **P****( X) = Pr(g(X))** is nonsensical, since they cannot be equal - the left-hand side is a statement in the meta-language, while the right-hand side is a formula of the system

*If * *X**then* *P*(*X*)

This, of course, is arrant nonsense, since **P****( X)** is an expression of the meta-language, then it cannot be proved within the system

*If * *X**then* *X*

which also serves to demonstrate the absurdity of Singh’s assertion.

The proof continues with a mish-mash of further errors. He goes on to state:

*Let* ** B_{1}(n)**,

But an enumeration is a function with one free variable, and here the free variable of the enumeration is represented by the subscripts **1**, **2**, **3**, … , so the general form of this enumeration is ** B_{m}(n)**, where m is the free variable and therefore n cannot be a free variable in the language of the enumeration. It is a variable of the system

Singh’s illogical subsequent treatment of the n as a free variable in B enables him to derive the expression:

(i) *B*_{k}(*n*) ↔ ¬*P*(*B*_{n}(*n*))

and then he states that by universal generalization, we have:

(ii) **∀ n(B_{k}(n) ↔ ¬P(B_{n}(n)))**

This is illogical nonsense, even if we ignore the fact as shown above that the system * N* cannot prove anything regarding

It is only by this illogical treatment of the n as a free variable in B that Singh manages to derive the expression:

** A ↔ ¬P(A)** (i.e.,

To be fair, Singh then states that he will now give what he calls a formal proof of this claim (but that hardly excuses the use of logically absurd statements such as treating a non-variable term as a variable). He goes on to state that:

*Let the ‘diagonalization’ of B*

**∃****x****(****B****(****x****) ∧ (****x = g****(****B****(****x****))))***.*

*Since g is a computable function, the relation*

* diag*(

** n** is the Gödel number of the diagonalization of the formula having exactly one free variable with Gödel number m

*is recursive and hence representable in N by some binary predicate, say,*

This is an elementary error. A computable function can refer to data that are not numbers (such as string data), and in fact the function g must refer to strings that are not numbers - symbol strings of the system * N*. But Singh has defined his system

Unfortunately, this paper that is replete with so many elementary logical errors is not an isolated example - it appears to be the case that many logicians and mathematicians are so accustomed to manipulating symbols within a single language that they seldom have to consider different levels of language. Because of this, they fail to take into account that whenever you are dealing with different levels of language, you have to be scrupulously careful not to confuse one level of language with another when you are manipulating symbols.

Antti Valmari, a Professor at Tampere University has published what he calls a “*rather easy yet rigorous proof of a version of Gödel’s first incompleteness theorem*” on the Arxiv website, see A Simple Character String Proof of the “True but Unprovable” Version of Gödel’s First Incompleteness Theorem.

Valmari bases his proof around a formal system that can make propositions about strings of symbols. He defines what symbols constitute the alphabet of the language of this system. A variable in this language is a string that begins with a lower case letter (a- z) followed by zero or more digits (0-9), e.g., b237 is a variable in this system.

Now, for a formal system to be able to prove anything:

- at least one axiom must be defined for the system
- at least one rule of inference must be defined for the system

However, Valmari fails to define any axioms or rules of inference that apply to his formal system. The closest he comes to providing such is when he makes remarks regarding the = symbol of the language. In essence he states that "X""Y" = "Z" is a ‘true’ proposition of the system if and only if XY is precisely the same symbol string as Z, and where "X""Y" is the concatenation of the strings "X" and "Y", and XY is the concatenation of the strings X and Y. (Valmari’s actual wording is: *"theorem"="theo""rem" is a true atomic proposition, and "theorem"≠"theo""rem" is not*.)

So, for the system to be able to prove, for example, "A""B" = "B", for any given symbol strings A, B, C, it must prove that AB is identical to C. This means that the system must be able to reference ** any** symbol string of the system, since A, B and C can represent

Herein lies a fundamental problem. It is impossible for a variable of a language to have the domain of * all* symbol strings of the language,since then it would include itself as a member of its own domain. This is a logical absurdity, since then it would be at the same time a variable, and a value that is substituted for a variable. Valmari seems to be completely unaware of this problem, and provides no argument to show how his nonchalant informal use of

The result is that he has given an informal description of a formal system that has no apparent means of proving even one proposition of that system. So, as Valmari has described it, his formal system is fundamentally incomplete anyway, because it cannot prove any propositions at all! As such, the remainder of Valmari’s incompleteness ‘proof’ is utterly worthless, since it is inapplicable to any useful formal system.

Another example of a flawed incompleteness proof is Gödel for Goldilocks: A Rigorous, Streamlined Proof of Gödel’s First Incompleteness Theorem, Requiring Minimal Background (v3) (PDF).

This critique addresses a revised version (version 3) of the paper posted on Arxiv by Dan Gusfield.

Gusfield has modified his paper considerably in response to my original critique of his paper (Gödel for Goldilocks (v1) (PDF)). My original critique is included Critique of Gusfield’s original paper version 1: below.

In his Section 2, Gusfield says that, given all computer programs of a given language that compute some function, we can “*conceptually order the strings representing the programs that compute the functions in A into a list L*”. Since Gusfield is claiming that his proof is rigorous, we can ignore the appeal to conceptualization and consider what Gusfield is actually defining in his description of the list.

In the description, we find that each symbol string is associated by this list with a specific natural number, where the numbers start at 1 and include all natural numbers. This is a straightforward definition of a function which has one free variable with the domain of natural numbers and the range of the function is a set of symbol strings. Given some natural number, this list function returns the appropriate value, which is some symbol string that is a computer program. Gusfield states:

“*Let f_{i} denote the function in A that appears in position i in L′.”*

In other words, the list function is denoted by ** f**, and the free variable of the list function is

Having defined this list function, Gusfield states that we can now define another function in terms of this list function ** f**, as follows:

“*Next, we define the function ¯ from the positive integers to {0, 1} as*

** f ¯( i ) = 1 − f_{i} ( i )**”

and Gusfield claims that this is a well-defined function (See his Section 4: Back to Gödel). He also asserts that the ** f ¯** function is not computable.

Since ** f ¯** is defined in terms of the list function and it is well-defined, it must be the case that the

But, according to Gusfield, when the free variable of his list function has been substituted by some specific value, another free variable suddenly springs up. Isn’t that convenient? Gusfield has defined a function ** f** with one free variable

In normal mathematics, given a list function whose range is symbol strings with one free variable, when that free variable is substituted by some specific number, the list function has a value of a string of symbols – but it cannot at the same time still have a symbol that is a free variable ** within the syntax of that language**. It can have symbols/symbol strings that are variables in whatever language is chosen for the computer programs, but the language of the list function is necessarily a language that is a meta-language to the language of the computer programs. The value returned by the list function is simply an object seen by the language, and which has no syntax within the language.

But in Gusfield’s definition, the value returned by the function is, ** at the same time**, a symbol string that has no syntax within the language, and

Gusfield asserts that his incompleteness proof applies to a language if it is a formal language that can express either ** f ¯( x ) = 1** or

- consistent,

and

- can prove the correct value of either
or*f*¯(*x*) = 1for every value of*f*¯(*x*) = 0.*x*

He then offers an argument that concludes that such a formal system cannot be both consistent and complete.

Well, since the ** f ¯** function is not computable, it’s questionable as to how any consistent formal system could prove a ‘correct’ value for

Having conceded that his original version was incorrect to claim that it applied to any formal system that “*can form any statement about integers*”, Gusfield has retreated to a position where his proof only applies to a bizarre non-conventional formal system. In his Introduction, Gusfield proclaims his proof is the “*just-right Goldilocks approach*” to an incompleteness proof. It is difficult to reconcile that assertion with the reality that his proof has no relevance to any conventional formal system.

Byunghan Kim is a university professor of mathematics and his paper, with the title “*Complete Proofs Of Gödel’s Incompleteness Theorems”*, is the basis of a lecture course to students. His paper consists of an impressive looking twenty-three pages with numerous definitions and equations replete with many symbols. In his Section “*Step 1: Representability of Recursive Functions in Q*” (page 9) Kim defines an function **n**_ (**n** underscored) as a string of symbols, where there are **n** of ** S** symbols, followed by a

Later on, in the Section “*Step 2: Axiomatizable Complete Theories are Decidable*” (page 13) he calls a language “* reasonable*” if there is a function

But the only definition of the underscore function **n**_ is a definition that only applies for **n** having the domain of natural numbers, that is, **n** can only take values that are natural numbers. So how can you have a ** V** quantity of

A reader has suggested that the occurrence of an underscore in ** V_** may indicate that here Kim is introducing a new function using the same terminology as for the previously defined

If I was one of Kim’s students I would be questioning why I was attending university lectures to be taught this sort of nonsense. Kim’s definition of a “* reasonable*” language is a language where there is some variable in the language that can refer to all symbols of the language, including that variable itself - so that definition of a “

In summary, Kim’s appellation of “* reasonable*” would seem to be one of the worst misnomers ever. And his paper is just another nonsensical incompleteness proof where the flaw is hidden in page upon page of
symbols and equations that may look impressive at first glance, but it is just another case of flash over substance.

This is the content of my critique of Gusfield’s original paper, Gödel for Goldilocks (v1) (PDF)

Gusfield introduces his proof with the claim that he is about to give a complete, rigorous proof of incompleteness. In section 4 of his paper, Gusfield naïvely claims that his proof applies to any formal system that can express his ** f ¯** function - and if you are wondering what that might mean, Gusfield goes on to say that his proof applies to any formal system that “

However, one presumes that is not what Gusfield intended. In his proof, he claims that the ** f ¯** function he has previously defined is a function from integers to integers, that is, its domain is integers, and its range is integers - and that therefore a suitable formal system can express this

So perhaps Gusfield meant a formal system that is a purely number-theoretic system? But a purely number-theoretic system that can form statements about natural numbers cannot necessarily express a function whose range and domain are natural numbers, when its definition includes entities that are not natural numbers. Gusfield’s ** f ¯** function is defined in terms of entities other than natural numbers, and thus the function is not defined in number-theoretic terms. It is in fact defined in terms of a list of computer programs, where the list is defined by taking the programs in order of their length, and, for programs of the same length, by some criterion according to the symbols used for the programs (i.e., in a similar style to a dictionary listing). But Gusfield doesn’t give any alternative definition of this function

The vagueness of Gusfield’s exposition makes a mockery of his claim of “rigorous” in the title of his paper, since his proof is certainly not rigorous by any stretch of the imagination. Despite his claims to the contrary, his paper is worthless since it does not establish any result of any value.

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There is a new addition to the page Yet another flawed incompleteness proof, where Berto’s proof of incompleteness in his book *There’s something about Gödel* comes under scrutiny.

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).

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.

There is now a new page on Chaitin’s Constant (Chaitin’s Omega), which demonstrates that Chaitin has failed to prove that it is actually algorithmically irreducible.

Flawed proofs of the Diagonal Lemma by Panu Raatikainen and Vann McGee have been added to the Diagonal Lemma web page.

16th Mar 2015 Bishops Dancing with Pixies?

23rd Feb 2015 Artificial Intelligence

31 Mar 2015 Cranks and Crackpots

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:

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

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