# Axioms of set theory and equivalents of axiom of choice

A senior thesis by Farighon Abdul Rahim, Spring 2014 (link)

*Abstract*: Sets are all around us. A bag of potato chips, for instance, is a set containing certain number of individual chip’s that are its elements. University is another example of a set with students as its elements. By elements, we mean members. But sets should not be confused as to what they really are. A daughter of a blacksmith is an element of a set that contains her mother, father, and her siblings. Then this set is an element of a set that contains all the other families that live in the nearby town. So a set itself can be an element of a bigger set.

In mathematics, axiom is defined to be a rule or a statement that is accepted to be true regardless of having to prove it. In a sense, axioms are self evident. In set theory, we deal with sets. Each time we state an axiom, we will do so by considering sets. Example of the set containing the blacksmith family might make it seem as if sets are finite. In truth, they are not! The set containing all the natural numbers {1, 2, 3…} is an infinite set. Our main goal for this paper will be the discussion of Axiom of Choice (AC) and its equivalents.