Sunday, January 28, 2024

Undecidability and Theories of Everything with Claus Kiefer

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Recorded: 2023/08/07 Released: 2024/01/28

Jim talks with Claus Kiefer about his recent essay on the relationship between the Gödel's incompleteness theoerems and the possibility of developing a theory of everything. Incompleteness was originally developed to show that every axiomatic system that is sufficiently robust admits well-formed statements that have a liar's paradox-like structure - if you assume the statement is true, you can prove it's false, and vice-versa. This statement is then said to be undecidable. Undecidability also famously comes up in the halting problem of computer science and the continuum hypothesis. Professor Kiefer speculates here that theories of everything are similarly undecidable.
------------------------------------------- Notes:

1. The article that we discussed in this program:
  • Kiefer, Claus, "Gödel's undecidability theorems and the search for a theory of everything" (2023) [arXiv]
2. Other papers referred to in this podcast:
  • Cubitt, T.,D. Perez-Garcia and H.M. Wolf, "Undecidability of the Spectral Gap." Nature 528 207 (2015) [arXiv]
  • Goedel,, K., "On Formally Undecidable Propositions of Principia Mathematica and Related Systems." Monatshefte für Mathematik und Physik 38 173 (1931) [Free]
    • The Undecidable [Amazon], M. Davis, ed. Reprints Goedel's paper and other work from the 1930's. Dover book.
3. Related Episodes of Physics Frontiers:
4. Books mentioned:
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  1. I worry that listeners will get the impression there is some intrinsic link between the continuum and incompleteness. There isn't: the continuum hypothesis is just a nice example of incompleteness. Discrete systems can just as well be incomplete: Turing machines, for example. And you can have uncountable models of your theory while remaining complete: Euclidean geometry or the first -order theory of the reals, for example.

    What's happening with Godels incompleteness is that the axioms can not single out one particular model. In the case of set theory, there are models of the axioms in which the continuum hypothesis is true, and ones where it is not. Since we can only prove what is true in every model (Godels completeness theorem), and disprove what is false in every model, we can neither oroce nor disprove the continuum hypothesis.

    1. Nice comment!

      I think that Claus' thesis is that there's a formal similarity between theories of everything and the continuum hypothesis. The problem isn't with continua, themeselves, but deciding on relationships between the sizes of higher order infinities.

      This is why we talked about the spectral gap and the continuum hypothesis instead of Goedel's theorems themselves.

  2. Maybe I was a bit quick to post the above. I'm certainly not accusing Kiefer of saying anything incorrect. You can make certain links between infinities and incompleteness. But it's all quite subtle.