MOPA

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We construct a theory definitionally equivalent to first-order Peano arithmetic PA and a non-standard computable model of this theory. The same technique allows us to construct a theory definitionally equivalent to Zermelo-Fraenkel set theory ZF that has a computable model. See my preprint https://arxiv.org/abs/2209.00967 for more details.

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It is well known that every countable recursively saturated model of $\mathsf{PA}$ has a full compositional truth predicate; that is, such a model is expandable to the theory $\mathsf{CT}^-$. It is also well known that such a truth predicate need not be inductive, or indeed, need not satisfy even $\Delta_0$ induction. Recently, Enayat and Pakhomov showed that $\Delta_0$ induction for the truth predicate is equivalent to the principle of disjunctive correctness: the assertion that for any sequence of sentences $\langle \phi_i : i \lt c \rangle$, the disjunction $\bigvee\limits_{i \lt c} \phi_i$ is evaluated as true if and only if there is $i \lt c$ such that $\phi_i$ is evaluated as true. In the absence of $\Delta_0$ induction, various pathologies can occur, including models of $\mathsf{CT}^-$ for which all nonstandard length disjunctions are evaluated as true. In this talk, we classify the sets X for which there is a model of $\mathsf{CT}^-$ in which X is exactly the set of those c such that the disjunctions of length c of 0 = 1 is evaluated as false. In particular, we see that X can be $\omega$ if and only if $\omega$ is a strong cut, and therefore the 'disjunctively trivial' models mentioned before are in fact arithmetically saturated. This is joint work (in progress) with Mateusz Łełyk, drawing heavily on unpublished work by Jim Schmerl.

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