#### 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.

#### Access

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