Georg Cantor Born March 3, 1845, Died January 6, 1918
Georg Ferdinand Ludwig Philipp Cantor is widely known for discovering transfinite numbers and for the creation of Set Theory. He is considered by most to be the greatest mathematical genius regarding infinite sets and concepts related to infinity.
It is my contention that the axioms of the real number plane alone do not accurately represents nature. There is an innate problem with a system based on the gain and debt of individual things which cannot symbolize the whole, since the fundamental axiom represents a nothing beyond which things exist. We have known in physics for quite a long time that there is no physical phenomenon of nothingness. Einstein revealed that both space and matter, and space and time, are one; or of the same element, and the concept of an empty space which is a nothing is invalid. Yet what symbol in mathematics represents a oneness that is and contains the whole of physics or cosmology. It is because we see or conceptualize the world as a quantity or grouping of individual things that we define infinity as a process, and such a process at any given real and definite existence remains finite.
The solution I think is radical as I explain here in this website, although I wouldn't propose foolishly that we abandon ordinary mathematical axioms, or even that they do not define an aspect of reality, however, it is possible to describe the universe mathematically with a primary system of values that exist inside of an ultimate whole value. This enlightening switch to values defines all things, physical and conceptual, as interior and less than the whole universe/existence, rather than all things as exterior and arisen above a primordial nothing/void.
Links that describe Set Theory