# Scott ranks and intended models

## Dariusz Kalociński

Certain mathematical theories are about a single, so-called intended, structure (e.g., arithmetic is about natural numbers) while others investigate general properties of all structures they axiomatize (e.g., group theory). The former theories are known as non-algebraic, while the latter as algebraic. One of the problems in the philosophy of mathematics concerns a systematic explanation of this phenomenon, ideally providing a theoretical notion that could explicate the concept of intendedness. I will briefly review some of the existing philosophical approaches regarding arithmetic, including the work of Halbach and Horsten [1], Button and Smith [2] and Walter Dean [3], among others. In the main part of my talk I will suggest a novel perspective on this problem, based on the measures of complexity of models, such as Scott ranks. I will try to explain basic technicalities involved in this notion and illustrate how it deals with the problem at hand with a few examples of algebraic as well as non-algebraic theories, including PA (by leveraging results from [4]) and weaker systems like Robinson’s or Presburger’s arithmetic.

## Bibliography

- Volker Halbach and Leon Horsten,
*Computational Structuralism*,,vol. 13 (2005), no. 2, pp. 174–186.*Philosophia Mathematica* - Tim Button and Peter Smith,
*The philosophical significance of {Tennenbaum}’s theorem*,,vol. 20 (2012), no. 1, pp. 114–121.*Philosophia Mathematica* - Walter Dean,
*Models and computability*,,vol. 22 (2014), no. 2, pp. 143–166.*Philosophia Mathematica* - Antonio Montalbán and Dino Rossegger,
*The structural complexity of models of arithmetic*,,(2023), pp. 1–17.*The Journal of Symbolic Logic*