University of Massachusetts, Dartmouth
Computer Oriented Mathematics
Mentor: Dr. Richard A. Zang, Associate Professor of Mathematics
A Survey of Infinity from the Greeks to the present: with emphasis on Georg Cantor's discovery of transfinite numbers
In modern mathematics the concept of infinity is used as if it were something that was proven to exist. Considering that it is still not determined whether our universe is bounded or unbounded and that there are still debates on the existence of an indivisible particle, it is prudent to shed some light on the subject. This study starts with the great Greek philosophers from 2500 years ago, pausing for significant review of Cantor’s work on transfinite numbers (commencing at the end of the 19th century), before finishing with a review of where we are at the present. This survey investigates the most influential mathematical views on the concept of infinity, where its size, existence, and usefulness are questioned throughout time. However, the study is focused on explaining how Georg Cantor, who is best known as the father of set theory, used that same theory to revolutionize the way we think of infinity-from being just an abstract concept to becoming a new class of actual numbers, the transfinite numbers. Lastly, the study analyzes the present debate about the existence and the usage of infinity in modern mathematics since its physical existence is still in question. In addition, the study explains why mathematics, as a both physical and philosophical science will continue to use infinity whether or not one can actually prove or disprove its physical existence.