What did cantor do?

Cantor revolutionised the mathematical concept of infinity. He lifted infinity from the realms of philosophy and theology and handed it to mathematicians and scientists. He identified infinities that were "countable" and "uncountable"; that is, infinities that could be put in one-to-one correspondence with the list of natural numbers 1,2,3,4,5,6… , and infinities that could not. So, for example, the even numbers are countably infinite, so are all the odd numbers.

Cantor defined all countable infinities as being the same size. This seems to introduce something of a paradox, because the list of natural numbers is infinite, and the list of odd numbers is infinite, intuition would suggest that one of these must be half the size of the other. We know that intuition is not always our best guide, so we should put aside intuition and look at how we can claim that these two infinities are the same size. the obvious answer is that we can go on putting one into one-to-one correspondence with the other for ever and we will not run out of either; but is that what infinity is really about? We can do this in theory, but never in practice. These infinities, undoubtedly, have their uses in the more esoteric realms of set theory and other branches of mathematics, but are of very dubious value when applied to the real world. One of the major problems with trying to integrate general relativity and quantum theory is that the equations of one, when applied to the other tend to lead to infinities, so the equations become nonsense.

Initially, the mathematics community was not over enthusiastic about Cantor's work, but after a time, mathematicians said "cool", or "hoc frigidulum est" or whatever the expression of the time might have been. Since then, Cantor has been widely quoted as having established that infinity was mathematically manipulatable.

What did Cantor really do to infinities? He discovered that there were ways of making infinities manageable by mathematicians. However, even he accepts that this is only a partial victory.

Barrow says: “Cantor’s most dramatic discovery was that infinities are not only uncountable, they are insuperable. He discovered that a never-ending ascending hierarchy of infinities must exist. There is no biggest of all that can contain them all. There is no Universe of universes that we can write down and capture."

Cantor called this "Absolute infinity", he likened it to "God", but at the same time established that this absolute infinity did not exist. Here is another paradox. If we argue that "There is no Universe of universes that we can write down and capture." There is no greatest infinity. Then, surely, we must argue that our so-called infinite series are not infinite, because they lead to no infinity "that we can write down and capture". They are unbounded, because we can neither see nor imagine an end, but we cannot say that they are infinite in any sense other than mathematical.