Logic and Language

Logic and Language

Copyright © James R Meyer 2012 - 2018 https://www.jamesrmeyer.com

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

Arindama Singh is a professor in the Department of Mathematics in the Indian Institute of Technology Madras and has written *Fundamentals 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.

*Other obviously flawed incompleteness proofs can be seen at:*

An Incompleteness Proof by Bernd Buldt

An Incompleteness Proof by Francesco Berto

An Incompleteness Proof by Dan Gusfield

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**

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**

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.

*Also see Errors in incompleteness proofs and Analysis of incompleteness proofs.*

*Other obviously flawed incompleteness proofs can be seen at:*

An Incompleteness Proof by Bernd Buldt

An Incompleteness Proof by Francesco Berto

An Incompleteness Proof by Dan Gusfield

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?.

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.

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

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.

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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Copyright © James R Meyer 2012 - 2018

https://www.jamesrmeyer.com