Personality prfofile #2 | Georg Cantor

Cantor and his theory of sets

“The essence of mathematics lies in its freedom”

Georg Cantor (1845-1918).

The development of cantor’s set theory:

Georg cantor was a german mathematician who in the late 19th century, funded the theory of sets.
To best understand Cantor's theory, I would like to give a little background to nineteenth century mathematics. Until the end of the nineteenth century no mathematician had managed to describe the infinite, beyond the idea that it is an absolutely unattainable value.
However,towards the late 19th century, the German mathematician, Weierstrass, gave a mathematically and logical solid definition of a limit and it is his definition of a limit that we use today and on which the calculus is founded.
But, as often happens, the resolution of one problem drew attention to another problem. It turned out that defining a limit necessitated a thorough definition of the real numbers, which in turn led to the study of infinite sets by Cantor.

His major work:

The development of cantor’s set theory made him the first to fully address the abstract concept of infinity. Cantor created modern set theory. He established the importance of one-to one correspondences between sets and founded the theory of transfinite numbers, the mathematics of the size of sets of different sizes. He showed, in particular, that infinities can have different sizes.

The Reception of Cantor’s work:

Cantor’s work was very controversial because it raised really difficult questions and many powerful and established mathematicians were critical of what he had achieved. Only by the end of his career he had achieved recognition and prizes but there were problems with his set theory which he could not resolve, for example Russell’s paradox involving the set of all sets that were not members of themselves. Is it a member of itself or not?
Cantor’s approach was to regard such collections as being too large and not really sets at all or as he put it inconsistent aggregates. Other mathematicians began to axiomatise set theory to try to exclude the possibility of contradictions.
But this new set of questions that arose form Cantor’s work should not take away from his great achievement of enabling us all to deal with the concept of the infinite in a careful, thoughtful and consistent manner.

Let me leave the final words to Cantor:

“My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things.”

Grorg Cantor