## File(s) under permanent embargo

# Arithmetical investigations: a study of models of arithmetic and purity of methods

thesis

posted on 2003-07-16, 00:00 authored by Andrew Peter AranaIn the mathematical part, we focus on computability-theoretic issues concerning models of first-order Peano arithmetic (PA). In Chapter 2, we investigate the complexity of m-diagrams of models of various completions of PA. We obtain characterizations that extend Solovay's results for open diagrams of models of completions of PA. In Chapter 3, we characterize sequences of Turing degrees that occur as {deg(T cap Sigma_n: ninomega}, where T is a completion of PA. In Chapter 4, we answer three questions asked by J. Knight concerning potential simplifications to Solovay's results. We show that these simplifications cannot be made, by proving some new independence results. In Chapter 5, we extend those independence results, using methods from higher recursion theory. In the philosophical part, we focus on purity constraints in mathematics. A proof is pure, roughly, if it uses methods `close' or `akin' to the statement being proved. We consider three different types of purity, which we call systematic, elementary and cognitive purity. Systematic purity, which we study in Chapters 7 and 8, has roots in Aristotle's views concerning scientific knowledge. Elementary purity, which we study in Chapters 9 through 12, has roots in Pappus' work in geometry, in Descartes' work in both geometry and epistemology, and in Hilbert's foundational work. Cognitive purity, which we discuss in Chapter 13, has roots in Kant's distinctions between philosophical and mathematical cognition, and between different sources of knowledge. We explain in detail what are each of the types of purity, consider what epistemic benefits are conferred by restricting ourselves to pure proofs, and discuss the consequences of apparent violations of these constraints in mathematical practice.

## History

## Date Created

2003-07-16## Date Modified

2018-10-05## Defense Date

2003-07-12## Research Director(s)

Peter Cholak## Committee Members

Peter Cholak Steven Buechler Patricia Blanchette## Degree

- Doctor of Philosophy

## Degree Level

- Doctoral Dissertation

## Language

- English

## Alternate Identifier

etd-07162003-140908## Publisher

University of Notre Dame## Program Name

- Mathematics

## Usage metrics

## Categories

No categories selected## Keywords

## Licence

## Exports

RefWorks

BibTeX

Ref. manager

Endnote

DataCite

NLM

DC