Abstract

In A Stroll Through Cantor’s Paradise: Appraising the Semantics of Transfinite Numbers, we confront the mysterious role of infinite (i.e., transfinite) numbers in mathematics. Our analysis of the subject is divided into three chapters. In the first chapter, “Building the Concepts of Set Theory”, we lay out the Zermelo-Fraenkel axioms and show how they allow for us to express the content of finite arithmetic in the language of set theory. These axioms are then augmented with several others to give foundation for the concept of ordinality, which is used to extend the linear number line into the transfinite realm.

In chapter two, “Goodstein’s Theorem”, we define the Goodstein sequence and invoke transfinite ordinal numbers to prove that every Goodstein sequence eventually reaches zero. We then discuss The Kirby-Paris Theorem, which shows that a system of higher-order arithmetic is needed to prove Goodstein’s Theorem.

In chapter three, “The Philosophy of Set Theory”, we consider the requisites for a formal theory to succeed in advancing our theoretical knowledge. We argue, a ́ la Immanuel Kant, that only the mathematical concepts that can be presented in intuition could tell us anything about the natural world. This implies that it is not justifiable to grant semantics to infinite numbers. Defining transfinite ordinals in a system of higher-order arithmetic fundamentally requires a bridge principle from the concepts of higher-order logic. We find, however, that none of these principles are sufficient grounds on which to do so.

Advisor

Zindulka, Ondrej

Second Advisor

Hustwit, Ronald E.

Department

Mathematics; Philosophy

Disciplines

Logic and Foundations | Mathematics | Set Theory

Publication Date

2017

Degree Granted

Bachelor of Arts

Document Type

Senior Independent Study Thesis

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© Copyright 2017 Matthew R. Buranosky