Is Prof. Claus Kiefer's application of Gödel (+ infinity) justified?

[nomadreid]

TL;DR

Prof. Kiefer applies Gödel and infinities in a suspicious manner in an ArXiv article (reference given); if anyone is familiar with his work, am I judging too hastily?

I have just looked through
https://arxiv.org/abs/2305.07331
by the physicist Prof. Dr. Claus Kiefer (Institut für Theoretische Physik; Universität zu Köln, Germany) .

The reason I am putting this in the logic thread instead of a physics thread is as follows. It appears to me that the author is treating Gödel's Incompleteness Theorems and also questions of infinity in a rather superficial manner, and hence his application of these concepts to proposed theories of quantum gravity appear to be to be suspect. My first reaction is, "ah, here we go again, a famous physicist delving outside his realm and misusing mathematical logic to confuse the ontological with the epistemological" …. that is, I am reminded of Sir Roger Penrose's ill-fated attempt (following Lucas) to use Gödel's First Incompleteness Theorem, and I have rarely found physicists interested in large cardinals. (Tellingly, Penrose's popular books containing his fallacy figure in the list of references for this ArXiv article.) However, my second reaction is that first impressions are often wrong, and I am always happy to learn from being wrong, so I looked for peer-reviewed articles on this theme by this author. Alas, I could not find any (although my only resource is those that are freely available on the Internet. At most I found a similar article by him in the May "Pour la science" (French version of "Scientific American") that was even less enlightening.) However, the article does give several references that I tried to look up, but the relevant ones (that is, other authors who, according to Prof. Kiefer, come to similar conclusions) are in journals with strong paywalls, and the abstracts do not seem promising.

Therefore, my request to anyone who is familiar with this physicist's work: I would appreciate an evaluation of his thesis as reflected in the ArXiv article.

Any guidance would be highly appreciated. Many thanks in advance.

 

[Demystifier]

Yes, the paper is very superficial, at several levels.

First, even if the number of degrees of freedom is finite, one still needs infinite sets. Even for a single degree of freedom $x$, the variable $x$ takes values from an infinite set. Thus the Godel theorem potentially may be relevant even for a universe with a finite number of degrees of freedom.

Second, dealing with infinite sets does not automatically imply that the Godel theorem is relevant. In particular, integer and real numbers can be axiomatized in a form that does not contain undecidable statements. See about Presburger arithmetic and Tarski axiomatization.

Third, just because a theory contains undecidable statements does not mean that it is not a final physical theory. A typical undecidable statement is of the self-referential form "This statement can't be proved", which hardly bothers any physicist as a signature that something of physical relevance is missing.

Fourth, even more interesting undecidable statements do not necessarily imply that something of physical relevance is missing. For example, an undecidable statement may be a general statement about any initial condition for a set of differential equations, while a physicist may only be interested in one particular initial condition, the one realized in the actual universe.

So yes, this is just one more abuse of the Godel theorem. In that context, I strongly recommend
https://www.amazon.com/Gödels-Theorem-Incomplete-Guide-Abuse/dp/1138427268?tag=pfamazon01-20

 

[nomadreid]

Super! Excellent and full explanation, thank you very much, Demystifier. Points I myself would not have come up with, and will follow up on. And thank you for the book recommendation: yes, I have seen the book sometime in the past, but I will dig it out and reread it.