Abstract

In this thesis we bind together two threads, model theory and the philosophy of mathematics, to argue that either mathematical claims have determinant truth value \textit{or} mathematical notions can be fully expressed by formal systems, but not both. These two beliefs are inconsistent with each other.

Model theory is a language which allows us to discuss the similarities between disparate mathematical systems as well as investigate which theories these systems satisfy. We will use model theory as a proverbial cannon to launch our broader argument. Specifically, we will set up the foundations of model theory before moving on to proving two key results: the Compactness Theorem and the Lowenheim-Skolem Theorems. It is this last one which forms a core part of our claim.

By establishing the existence of non-isomorphic models, the Lowenheim-Skolem Theorems pose a question for determinate truth which cannot be answered using only formal methods. Thus, the Lowenheim-Skolem Theorems, determinate truth, and formalism form an inconsistent triad. Furthermore, any attempt to reject the Lowenheim-Skolem Theorems must also reject formalism. Thus, if the Theorems are true, either determinate truth or formalism is false. If the Theorems are false, then determinate truth comes at the cost of formalism. Stuck between these two options, we must admit that mathematical claims having determinate truth value and mathematical notions being fully expressed by a formal system are inconsistent with each other.

Advisor

Bowen, Jennifer

Second Advisor

Thomson, Garrett

Department

Mathematics; Philosophy

Disciplines

Logic and Foundations | Logic and Foundations of Mathematics | Set Theory

Publication Date

2022

Degree Granted

Bachelor of Arts

Document Type

Senior Independent Study Thesis

Download

Share

COinS
 

© Copyright 2022 Brendan Dufty