Logic and
Load the menuLoad the menu

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 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.

Interested in supporting this site?

You can help by sharing the site with others. You can also donate at Go Get Funding: Logic and Language where there are full details.



As site owner I reserve the right to keep my comments sections as I deem appropriate. I do not use that right to unfairly censor valid criticism. My reasons for deleting or editing comments do not include deleting a comment because it disagrees with what is on my website. Reasons for exclusion include:
Frivolous, irrelevant comments.
Comments devoid of logical basis.
Derogatory comments.
Long-winded comments.
Comments with excessive number of different points.
Questions about matters that do not relate to the page they post on. Such posts are not comments.
Comments with a substantial amount of mathematical terms not properly formatted will not be published unless a file (such as doc, tex, pdf) is simultaneously emailed to me, and where the mathematical terms are correctly formatted.

Reasons for deleting comments of certain users:
Bulk posting of comments in a short space of time, often on several different pages, and which are not simply part of an ongoing discussion. Multiple anonymous user names for one person.
Users, who, when shown their point is wrong, immediately claim that they just wrote it incorrectly and rewrite it again - still erroneously, or else attack something else on my site - erroneously. After the first few instances, further posts are deleted.
Users who make persistent erroneous attacks in a scatter-gun attempt to try to find some error in what I write on this site. After the first few instances, further posts are deleted.

Difficulties in understanding the site content are usually best addressed by contacting me by e-mail.


Based on HashOver Comment System by Jacob Barkdull

Copyright   James R Meyer   2012 - 2024