March 15
Michał Godziszewski,
University of Warsaw
Tennebaum's Theorem for quotient presentations and model-theoretic skepticism
A computable quotient presentation of a mathematical structure $\mathcal A$ consists of a computable structure on the natural numbers $\langle \mathbb{N},\star,\ast,\dots \rangle$, meaning that the operations and relations of the structure are computable, and an equivalence relation $E$ on $\mathbb{N}$, not necessarily computable but which is a congruence with respect to this structure, such that the quotient $\langle \mathbb{N},\star,\ast,\dots \rangle$ is isomorphic to the given structure $\mathcal{A}$. Thus, one may consider computable quotient presentations of graphs, groups, orders, rings and so on. A natural question asked by B. Khoussainov in 2016, is if the Tennenbaum Thoerem extends to the context of computable presentations of nonstandard models of arithmetic. In a joint work with J.D. Hamkins we have proved that no nonstandard model of arithmetic admits a computable quotient presentation by a computably enumerable equivalence relation on the natural numbers. However, as it happens, there exists a nonstandard model of arithmetic admitting a computable quotient presentation by a co-c.e. equivalence relation. Actually, there are infinitely many of those. The idea of the proof consists is simulating the Henkin construction via finite injury priority argument. What is quite surprising, the construction works (i.e. injury lemma holds) by Hilbert's Basis Theorem. The latter argument is joint work with T. Slaman and L. Harrington.