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Often, people who study mathematics learn theorems to prove results in and about the vast array of branches of mathematics (Algebra, Analysis, Topology, Geometry, Combinatorics, etc.). This helps them move forward in their understanding; but few ever question the basis for these theorems or whether those foundations are sucient or even secure. Theorems come from our foundations of mathematics, Axioms, Logic and Set Theory. In the early20th century, mathematicians set out to formalize the methods, operations and techniques people were assuming. In other words, they were formulating axioms. The most common axiomatic system is known as the Zermelo-Fraenkel axioms with the addition of the Axiom of Choice (AC), although AC is still a controversial axiom. These axioms helped build our notion of infinity, which turns out to be much more complicated than what had been suspected. Earlier mathematicians from the Greeks to Gauss even refused to view infinity as an actuality, and only referred to “potential infinities”. Now our axioms allow multiple sizes of infinity. With the help of ordinals and cardinals, we can start to see a framework for these previously obscure notions. This leads to the even more difficult question: what exactly are the real numbers? The Continuum Hypothesis provides one approach, yet the full story remains “unsolved” by mathematicians to this day.



Thesis Comittee

Dr. Ward Heilman, Thesis Advisor
Dr. John Pike, Committee Member
Dr. Rachel Stahl, Committee Member

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Original document was submitted as an Honors Program requirement. Copyright is held by the author.