Axiomatic Set Theory

Course Description

A development of Zermelo-Fraenkel set theory (ZF), including the axiom of choice (ZFC), a system in which all of mathematics can be formulated (i.e., entails all theorems of mathematics). Although, we’ll do an exposure to transfinite ordinals and cardinals in general so that you can get a sense for how stupendously “large” these can be, the main thrust concerns certain simple, seemingly well-posed conjectures whose status appears problematic. For example, the Continuum Hypothesis (CH) is the conjecture that the cardinality of the real numbers is the first uncountable cardinality, i.e., the first cardinality greater than that of the set of natural numbers. Equivalently, there is no uncountable subset of real numbers strictly smaller in cardinality than the full set of reals. (You’d think that if there were one, you would be able eventually to find such.) Cantor thought that CH is true, but could not prove it. Gödel showed, at least, that if ZFC is consistent, then so is ZFC+CH. However, Paul Cohen later proved that if ZFC is consistent, then so is ZFC + the negation of CH. In fact, CH could fail in astoundingly many ways. For example, the cardinality of the continuum could be (weakly) inaccessible, i.e., of a cardinality that cannot even be proved to exist in ZFC (although the reals can certainly can be proved to exist in ZFC). So, are there further, intuitively true axioms that can be added to ZFC to resolve the cardinality of the continuum, and CH is definitely true or false? Or, as Cohen thought, does CH simply lack a definite truth value?

Code

AS.150.422

Credits

3

Subject

Philosophy

Programs That Include This Course

• Bachelor's Mathematics, Bachelor of Arts
• Bachelor's Philosophy, Bachelor of Arts