Oral: Can Wilkie's Theorem serve as a foundation for Computable Physics?

There exists strong philosophical arguments for why physics built upon continuous real-valued (or complex-valued) fields is unsatisfactory[1], but I argue that the negative result to Tsirelson's problem[2] implied by the seminal result $MIP^*=RE$[3] indicates that the discussion is not a philosophical one, but has physical implications. I demonstrate that if Tegmark's Computable Universe Hypothesis is true, then Turing's definition of computability[4] is trivially inadequate given the dimensionality of space-time. The basic physical requirements of a mathematical framework for building physics will be outlined, and one possible path forward by expanding the definition of computability to exponention is proposed which is motivated by Wilkie's Theorem[5] from model theory.

1. Tegmark, M. (2007). The Mathematical Universe. Foundations of Physics, 38(2), 101–150. https://doi.org/10.1007/s10701-007-9186-9

2. Scholz, V. B., & Werner, R. F. (2008). Tsirelson’s Problem.

3. Ji, Z., Natarajan, A., Vidick, T., Wright, J., & Yuen, H. (2022). MIP*=RE.

4. Turing, A.M. (1937), On Computable Numbers, with an Application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, s2-42: 230-265. https://doi.org/10.1112/plms/s2-42.1.230

5. A.J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted pfaffian functions and the exponential functions, J. Amer. Math. Soc. 9 (1996), pp. 1051–1094.