A senior thesis by Chloe Munroe, Spring 2019

 Abstract: The hyperreal number system $\ast\mathbb R$ forms an ordered field that contains $\mathbb R$ as a subfield as well as infinitely large and small numbers. A number is defined to be infinitely large if $x >n$ for all $n = 1, 2, 3, \ldots$ and infinitely small if $x <1/n$ for all $n = 1, 2, 3\ldots$ This number system is built out of the real number system analogous to Cantor’s construcion of $\mathbb R$ out of $\mathbb Q$. The new entities in $\ast\mathbb R$ and the relationship between the reals and hyperreals provides an appealing alternate approach to real (standard) analysis referred to as nonstandard analysis. This approach is based around that principle that if a property holds for all real numbers then it holds for all hypereal numbers, known as the transfer principle. By only using the fact that $\ast\mathbb R$ is an ordered field that has $\mathbb R$ as a subfield, includes unlimited numbers and satisfies the transfer principle the topics of analysis can be explored.