Tennenbaum’s Theorem and Non-Classical Arithmetic
Nuno Maia
Abstract:
This works aims to address the significance of Tennenbaum’s Theorem for the philosophy of model theory, from the perspective of non-classical inconsistent models of arithmetic. Several authors have recently argued that Tennenbaum’s Theorem, when coupled with the claim that intended addition is computable, is capable of isolating the intended models of Peano Arithmetic up to a single isomorphism type. Such argument, which we will call the argument from Tennenbaum’s Theorem, is particularly welcoming for a class of views in the foundations of mathematics that reject great epistemic access to mathematical objects. By focusing on a specific class of paraconsistent models of pa, as well as their features regarding cardinality and computability issues, we will argue that when pursued to its last consequences the argument from Tennenbaum’s Theorem leads to very unintuitive results. In fact, we will show that the insistence on the computability of the intended addition function leads to placing inconsistent models in the class of intended ones. We discuss how this is an unwanted result for the advocate of the argument from Tennenbaum’s Theorem and possible ways to block it. As we will see, the unexpected consequences are not so easily dismissed. We conclude that the argument from Tennenbaum’s Theorem is too weak to establish its conclusion.