Weak Fragments of Arithmetic

Author

Vina Maureen Castelli, Southern Illinois University CarbondaleFollow

Date of Award

8-1-2015

Degree Name

Master of Science

Department

Mathematics

First Advisor

Calvert, Wesley

Abstract

The purpose of this thesis is to investigate which weak fragments of arithmetic are essential to proving certain theorems that students are required to use, understand, and/or prove according to the Common Core Standards. The strength of a fragment sometimes correlates to computational complexity and can serve as a proxy for cognitive difficulty. To determine which are the weak fragments of arithmetic are essential to a proof, we first interpret the chosen standard into formal mathematical language; this process generates a theorem to be proven. We proceed to prove this theorem using weak fragments of arithmetic such as IOPEN or $I\Delta_0$. After we have proven a theorem, we must then show that we used the minimal fragment possible in our proof. This has only been done for one theorem, namely The Remainder Theorem. We found a proof for The Remainder Theorem using $I\Delta_0$. To prove that this is the minimal fragment needed for the proof (that IOPEN would not suffice), we built a non-standard model of IOPEN using forcing in which The Remainder Theorem is not true.