1

From this answer:

In particular, while PA is still overkill, there are theories of arithmetic much stronger than arithmetic with successor which are too weak for the Tennenbaum phenomenon to hold for them.

For which theories of arithmetic does Tennenbaum's theorem fail?

Carla only proves trivial prop
  • 704

1 Answers1

4

This is still an active area of research. D'Aquino's $1997$ article Towards the limits of the Tennenbaum phenomenon is a bit old at this point, but has an excellent introduction, and I believe the big open question mentioned there (whether Tennenbaum applies to $IE^-$) is still open. McAloon showed in $1982$ that Tennenbaum applies to $I\Delta_0$ (this is mentioned in D'Aquino's intro), and below $I\Delta_0$ everything becomes quite complicated.

More recently, Pakhomov has shown that the Tennenbaum phenomenon is surprisingly (to me at least) "language-specific" - see How to escape Tennenbaum's theorem.

Noah Schweber
  • 245,398
  • Regarding last paragraph: wow, that really is odd. Makes me think there is something I don't understand about this stuff as well as I thought I did (and I did not think I understand it that well to begin with...). – tomasz Oct 30 '22 at 10:52