Logic Seminar
Matt Foreman
UC Irvine
Hilbert's 10th problem for measure preserving transformations.
Abstract: In his seminal 1932 paper von Neumann asked whether it is possible to tell whether time is going forwards or time is going backward from the statistics of a measure preserving system. It was not until 1941 that Anzai produced the first example of a measure preserving system where $T$ is not isomorphic to $T^{-1}$.
In this talk I show that there is a one-to-one, primitive recursive map $F$ that maps Gödel numbers of $\Pi^0_1$-sentences to recursive, measure preserving, invertible, ergodic diffeomorphisms of the 2-torus such that:
\[\phi \mbox{ is true if and only if } F(\phi) \mbox{ is isomorphic to }F(\phi)^{-1}.\]
As corollaries there are non-isomorphic ergodic measure preserving diffeomorphism of the torus:
- $T_{RH}$ such that $T_{RH}\cong T_{RH}^{-1}$ if and only if the Riemann Hypothesis is true.
- $T_{GC}$ such that $T_{GC}\cong T_{GC}^{-1}$ if and only if Goldbach's Conjecture is true.
- $T_{ZFC}$ such that $T_{ZFC}\cong T_{ZFC}^{-1}$ if and only if ZFC is consistent.
- $T$ such that $T\cong T^{-1}$ but this is independent of ''ZFC + there is a supercompact cardinal."
Tuesday February 25, 2020 at 3:00 PM in 427 SEO