John Horton Conway discovered a proper class of numbers called surreal numbers. The surreal number field is a totally-ordered field that contains R and all of Cantor's ordinals. Within it we find infinite numbers as well as infinitesimal numbers. The class of all positive surreal numbers can be partitioned into convex subclass called commensurate classes, and these represent the different "sizes" of numbers. This thesis is an attempt to determine if it is possible to construct fractal objects within the surreal number field in a way that reflects their typical construction within the real number field. We give insight into the answer of this question by defining a surreal analogue to the Cantor set that we call a Cantor-Conway class. Our definition generalizes the base-three representation of the classical Cantor set. We also discuss a method of showing that each member of a particular subset of Cantor-Conway classes is equal to the union of the members of an increasing sequence (indexed by every ordinal number) of full sets, thus mimicking one of the standard constructions of the Cantor set. We think of this as starting with a "seed" set and then growing both "upwards into infinity" as well as "downwards" into the "tiny spaces" between previous points.
Call Number
LE3 .A278 2024
Date Issued
2024
Supervisor
Degree Name
Master of Science
Degree Level
Masters
Degree Discipline
Affiliation
Abstract
Publisher
Acadia University