O wain G riffin . On the Uniqueness and Determinacy of the Standard Model of Arithmetic . The Ohio State University. 2026. Supervised by Stewart Shapiro. MSC: 00A30, 03A05, 03A10. Keywords: philosophy of mathematics, non-standard models, model theory

The dissertation concerns the intended interpretation of mathematical theories. Reasonably straightforward results in model theory show that the axioms of our mathematical theories fail to determine their own intended interpretation, even up to isomorphism. Quite generally, any first-order mathematical theory purporting to describe a particular subject matter will have unintended, non-isomorphic models of its axioms. These unintended interpretations satisfy not only the axioms of the theory but indeed the complete set of truths concerning the domain. As a result, axiomatic theories cannot uniquely determine their mathematical content. However, this raises a worrying question. Simply put, if the relevant axioms could not determine the subject matter of a theory, what possibly could? In the dissertation, I explore this problem. To do so, I begin in chapter one by situating the project with respect to the broader philosophical literature, and utilise some basic techniques for the construction of non-standard models to introduce the general challenge with which the dissertation is concerned. In my second chapter, I show that model-theoretic underdetermination can entail a skeptical conclusion about mathematics which presents a substantial challenge for philosophers. In doing so I explore three prominent anti-skeptical arguments, and show that they are each subject to significant structural defects. I therefore conclude that the problem of unintended interpretations remains a significant problem for the philosophy of mathematics. In the third chapter, I draw attention to an important presupposition that has been widely overlooked throughout the debate: the presupposition of classical mathematics. In doing so, I utilise theorems of intuitionistic mathematics to show that intuitionistic arithmetic and set theory do not face the same problems as their classical counterparts. As a result I argue that, contrary to received wisdom, there is a stable view which can be maintained in light of the skeptical challenge which can also be seen as providing an independent argument for intuitionism. In my final chapter, I develop a novel proposal for a purely algebraic conception of mathematics which seeks to reframe the problem of unintended interpretations as a beneficial feature of modern axiomatic systems, rather than a defect. In doing so I argue that no mathematical theory possesses only a single uniquely correct interpretation. Rather, there are a plurality of structures corresponding to our theories, each of which should be regarded as legitimate. The dissertation concludes that there is significant value to be had both mathematically and philosophically from investigating unintended interpretations, and urges further research into the intersection of philosophy and model theory. Abstract prepared by Owain Griffin. Current address : 350 University Hall, 230 North Oval Mall, USA. E-mail: [email protected] . URL: https://etd.ohiolink.edu/acprod/odb_etd/r/etd/search/1?clear=0,1,5,10,20,21,1001 .