Logic and Language

Copyright   James R Meyer    2012 - 2023 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 Antti Valmari

Antti Valmari, a Professor at the University of Jyväskylä 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).

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:

1. at least one axiom must be defined for the system
2. 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" = "C", 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 any symbol string of the system.

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 " as a delimiter can be replicated within his system - a system that he says can make propositions about strings of symbols.

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.

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Copyright   James R Meyer   2012 - 2023
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