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:

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 (version 2 is discussed here).

*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

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.

*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