Digital Commons Network^™

On The Fundamental Group Of Plane Curve Complements, Mitchell Scofield

Boise State University Theses and Dissertations

Given a polynomial f(x,y) monic in y of degree d, we study the complement ℂ²-C, where C is the curve defined by the equation f(x,y)=0. The Zariski-Van Kampen theorem gives a presentation of the fundamental group of the complement ℂ²-C. Let NT be be the set of complex numbers x for which f(x,y) has multiple roots (as a polynomial in y). Let : ℂ − NT → ℂ^d − Δ be the map that …

Computable Reducibility Of Equivalence Relations, Marcello Gianni Krakoff

Boise State University Theses and Dissertations

Computable reducibility of equivalence relations is a tool to compare the complexity of equivalence relations on natural numbers. Its use is important to those doing Borel equivalence relation theory, computability theory, and computable structure theory. In this thesis, we compare many naturally occurring equivalence relations with respect to computable reducibility. We will then define a jump operator on equivalence relations and study proprieties of this operation and its iteration. We will then apply this new jump operation by studying its effect on the isomorphism relations of well-founded computable trees.

Formally Verifying Peano Arithmetic, Morgan Sinclaire

Boise State University Theses and Dissertations

This work is concerned with implementing Gentzen’s consistency proof in the Coq theorem prover.

In Chapter 1, we summarize the basic philosophical, historical, and mathematical background behind this theorem. This includes the philosophical motivation for attempting to prove the consistency of Peano arithmetic, which traces itself from the first attempted axiomatizations of mathematics to the maturation of Hilbert’s program. We introduce many of the basic concepts in mathematical logic along the way: first-order logic (FOL), Peano arithmetic (PA), primitive recursive arithmetic (PRA), Gödel's 2nd Incompleteness theorem, and the ordinals below ε₀.

In …