Browsing by Subject "axiom"

The history of set theory is a long and winding road. From its inception, set theory has grown to become its own flourishing branch of mathematics with a pivotal role in the attempt to establish a foundation for all of mathematics and as such its influence is felt in every corner of the mathematical world as it exists today. This foundational effort, in the form of establishing new set theoretic axioms, is still ongoing and a big driving force behind this movement is the many unanswered questions that remain out of reach of the set theory of today. One of the most well known of these open questions is that of the Continuum Hypothesis. In this thesis we will first dive into the history of set theory, starting by looking at the role that infinity has played in the history of mathematics. From the ancients Greeks to Cantor who finally brings infinity into mathematics in a major way through set theory. We look at the development of a foundation for mathematics through the axiomatization of set theory and then focus on the role the Continuum Hypothesis played in this effort, leading up to Gödel’s and Cohen’s proofs that showed its independence and beyond that to the research being done today. We then turn our attention to potential candidates for new axioms that would solve the Continuum Hypothesis. First we take a closer look at Gödel’s constructible universe, in which the Continuum Hypothesis is true. We look at how it is built and consider the potential results of accepting the corresponding Axiom of Constructibility as a new axiom of set theory. In the final section we examine Chris Freiling’s proposed Axioms of Symmetry, which imply the negation of the Continuum Hypothesis. After looking at Freiling’s constructions in detail we consider the arguments for and against accepting them as new axioms.