Simplest model properties for Peano Arithmetic: On a question of Montalban and Rossegger.
Mateusz Łełyk
Authors: Mateusz Łełyk and Patryk Szlufik
This is joint work with Patryk Szlufik from the University of Warsaw.
As famously shown by Scott, every countable structure can be characterized, up to isomorphism, by a sentence of infinitary language \(L_{\omega_1, \omega}\) which allows for conjunctions and disjunctions over arbitrary countable families of formulae (over finitely many variables). Formulae of this language can be naturally assigned ranks based on the number of alternations of existential connectives (disjunctions and existential quantifiers) with universal ones (conjunctions and universal quantifiers). This gives rise to a natural complexity measure for countable models: the Scott rank of a model \(\mathcal{M}\) is the least \(\alpha\) such that \(\mathcal{M}\) can be uniquely characterized by a sentence of rank \(\alpha+1\) (and starting from the universal quantifier). The developments of computable model theory witness that the Scott rank is a very robust notion integrating other well established tools from descriptive set theory. model theory and computability.
In “The Structural Complexity of Models of Arithmetic” Antonio Montalban and Dino Rossegger pioneered the Scott analysis of models of Peano Arithmetic. They characterized the Scott spectrum of completions PA , i.e. the set of ordinals which are Scott ranks of countable models of a given completion \(T\) of PA. A particulary intriguing outcome of their analysis is that PA has exacty one model of the least rank, the standard model, and the Scott rank of every other model is infinite. Additionally they studied the connections between Scott ranks and model-theoretical properties of models, such as recursive saturation and atomicity, rasising an open question: is there a non-atomic homogeneous model of PA of Scott rank \(\omega\)?
In the talk we answer the above question to the negative, showing that the nonstandard models of PA or rank \(\omega\) are exactly the nonstandard prime models. This witness another peculiar property of PA: not only it has the simplest model, but also every its completion has a unique model of the least Scott rank.