This page is keyboard accessible:
• Use Tab, Shift + Tab keys to traverse the main menu. To enter a sub-menu use the Right Arrow key. To leave a sub-menu use the Left Arrow or the Escape key.
• The Enter or the Space key opens the active menu item.
• To skip the menu and move to the main content, press Tab after the page loads to reveal a skip button.
• To get back to the top of the page anytime, press the Home key.
• For more information, click here: Accessibility   Close this tip.

Note: Full functionality of this web page requires JavaScript to be enabled in your browser.
 

Oh no ! Yet Another Flawed Incompleteness Proof

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

A Flawed Incompleteness Proof by Byunghan Kim

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 0 symbol, i.e., n_ = SS … S0, where there are n S’s in the sequence, e.g. 6_ = SSSSSS0.

 

Later on, in the Section “Step 2: Axiomatizable Complete Theories are Decidable” (page 13) he calls a language “reasonable” if there is a function h in the language where V_ = h(V) applies, where V is a variable of the language (along with other conditions). This means that the domain of V is all variables of the language - that is, any variable of the language may be substituted for V.

 

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 S’s if V can only be a variable but not a number? Answer - since V is a symbol that is a variable, not a number - you can’t.

 

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 n_, without stating so explicitly. If that is the case, that is extremely bad practice. But even if it is the case, Kim is still saying that a language is “reasonable” if it includes a function h where the free variable has a domain that includes all variable symbols of the language - but then that free variable of h has a domain that includes that variable itself, which is nonsensical. Furthermore, Kim asserts in his Fixed Point Theorem (Section: “Step 3: The Incompleteness Theorems and Other Results”) that since the function dg is recursive, it is representable in Peano arithmetic - but Kim’s prior argument was that a function is representable in Peano arithmetic if it is a recursive and if it is a number-theoretic function –the function dg is not number-theoretic, since it is defined in terms of the h function, whose free variable has a domain that includes symbols that are not natural numbers.

 

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 “reasonable” language would just be a definition of a self-referential language, which means that sentences of the language can refer to themselves. And it is not surprising that you will get paradoxes in such a language, in the same way that you can get a ‘liar’ paradox in a self-referential language like English. And if you are going to have a proof that applies to such languages, then you don’t need the vast amount of material that Kim uses to give a proof involving self-reference - all you have to do is apply something like the diagonal lemma, where your proof is only a page long. But it isn’t a proof that is going to apply to any logically valid formal system.

 

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.

section divider

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

An Incompleteness Proof by Arindama Singh

An Incompleteness Proof by Antti Valmari

section divider

 

 

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?.

section divider
 

The Lighter Side

NEWS

Paper on the diagonal proof

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.

section divider

Other recently added pages

The Platonist Rod paradox

 

The Balls in the Urn Paradox

 

How you can tell if someone is a crackpot

 

Platonism’s Logical Blunder

section divider

Lebesgue Measure

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

section divider

Easy Footnotes

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).

section divider

O’Connor’s “computer checked” proof

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.

Previous Blog Posts

Descartes’ Platonism

The duplicity of Mark Chu-Carroll

A John Searle Inanity

Man versus Machine

Fake News and Fake Mathematics

Ned Block’s Blockhead

Are we alone in the Universe?

Good Math, Bad Math?

Bishops Dancing with Pixies?

Artificial Intelligence

Cranks and Crackpots

The Chinese Room

Links

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:

Gödel Links

 

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

Gödel, Minds, and Machines

Printer Friendly

All pages on this website are printer friendly, and will print the main content in a convenient format. Note that the margins are set by your browser print settings.


Note: for some browsers JavaScript must be enabled for this to operate correctly.

Comments

Comments on this site are welcome, please see the comment section.

 

Please note that this web site, like any other is a collection of various statements. Not all of this web site is intended to be factual. Some of it is personal opinion or interpretation.

 

If you prefer to ask me directly about the material on this site, please send me an e-mail with your query, and I will attempt to reply promptly.

 

Feedback about site design would also be appreciated so that I can improve the site.


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