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:

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 a function **n**_ (**n** underscored) as a string of symbols, where there are **n** of ** S** symbols, followed by a

*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

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.

*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