Tagged: Georg Cantor

Continuum Hypothesis

Georg Cantor originally proposed that there is no infinite set with a cardinal number between that of the “small” infinite set of natural numbers (N) and the “large” infinite set of real numbers C (the “continuum”).  “There is no set whose cardinality is strictly between that of the integers and that of the real numbers.”  This means that N=Aleph0, C=Aleph1. Gödel showed that no contradiction arises if the continuum hypothesis is added to conventional Zermelo-Fraenkel set theory. However, Paul Cohen proved in 1963 that no contradiction arises if the negation of the continuum hypothesis is added to set theory. Together,...

A History of Set Theory

A set is a collection of things. A set can consist of numbers or letters (such as 1, 2, 3, 4 or a, b, c, d) or of objects (such as chairs or books). Set theory is the field of mathematics that deals with the properties of sets that are independent of the things that make up the set. For example, a mathematician might be interested in knowing about sets S and T without being interested at all about the concrete elements of the sets. The following article addresses the history of set theory, but it also serves as an...

Georg Cantor

from the MacTutor History of Mathematics archive, article byJ J O’Connor and E F Robertson:  “Georg Cantor‘s father, Georg Waldemar Cantor, was a successful merchant, working as a wholesaling agent in St Petersburg, then later as a broker in the St Petersburg Stock Exchange. Georg Waldemar Cantor was born in Denmark and he was a man with a deep love of culture and the arts. Georg’s mother, Maria Anna Böhm, was Russian and very musical. Certainly Georg inherited considerable musical and artistic talents from his parents being an outstanding violinist. Georg was brought up a Protestant, this being the religion...

Georg Cantor Quotes

The final problem which Cantor grappled with was the realization that there could be no set containing everything, since, given any set, there is always a larger set — its set of subsets. Furthermore, he believes that infinity actually exists – it is not just a mathematical construct.  Cantor came to the conclusion that there is an Absolute Infinite that transcends transfinite numbers. It has mathematical properties, and he identified this concept with God. Subsequently, he believed that his new mathematics is actually a form of theology.  Here are some quotes by Georg Cantor: I have never proceeded from any...