Date of Award
Bachelor of Science
numbers, set, y0, countable
At the heart of mathematics is the quest to ﬁnd patterns and order in some set of similar structures, whether these be shapes, functions, or even numbers themselves. In the late 1800’s, there was a strong focus in the mathematical community on the study of real numbers and sequences of real numbers. Mathematicians quickly realized, however, that in order to do any meaningful investigation into the properties of sequences of real numbers, they needed a better deﬁnition of real numbers than the loose intuitions that had been sucient for the generations prior. This led Georg Cantor (March 3, 1845 - January 6, 1918) to create his own deﬁnition of real numbers during his investigations into trigonometric series. As Cantor continued his work in formal deﬁnitions of number systems, he slowly realized that there was good reason to extend the real numbers beyond their traditional conﬁnes into what are now known as transﬁnite numbers. In doing this, Cantor set down a road, which, while revealing to him that some inﬁnite sets were bigger than others, eventually drove him to be institutionalized. In the discussion that follows, we will provide an exposition on the “sizes” of many familiar sets. In this context, we will use cardinality as our measure of size. Speciﬁcally, we will establish that N is isomorphic to Q, and thus the two have equal cardinality.
Warrener, Michael, "Cantor's Infinity" (2016). Honors Theses and Student Projects. 220.