Logic and
Language

Copyright   James R Meyer    2012 - 2024 https://www.jamesrmeyer.com

# 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 his 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.

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Copyright   James R Meyer   2012 - 2024
https://www.jamesrmeyer.com