Georg Cantor

I entertain no doubts as to the truths of the transfinites, which I recognized with God’s help and which, in their diversity, I have studied for more than twenty years; every year, and almost every day brings me further in this science. The disagreement of even some of the most distinguished mathematicians with my views does not shake my faith in them. I feel now as if I were standing at a great height and looking down on a confused crowd of opponents.

Mathematics is entirely free in its development, and its concepts are only limited by the logical rules.

In mathematics the art of proposing a question must be held of higher value than solving it.

I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author. Thus I believe that there is no part in reality more important than the actual infinite, and that as a matter of fact all of otherwise so-called absolute infinity is only a form of its reverberation, so that a real proper understanding of the actual infinite is just as important to mathematics as that of the finite, and is absolutely necessary to our understanding of the latter.

The transfinite numbers are in a certain sense themselves the absolute infinite.

Every transfinite consistent multiplicity, in itself, in all parts and in all respects, can be thought of as a whole only as a unity realized in the element of pure quantity (that is, as one or many, as a simple or a composite number, up to the concept of a real number in general).

The essence of mathematics lies in its freedom.

The infinite! No other question is it insisted upon so much.

A set is a gathering together into a whole of definite, distinct objects of our intuition or our thought.

The transfinite species are just as much at the disposal of the intentions of the Creator and His absolute power, as the finite.

I realize that in this undertaking I place myself at a disadvantage compared to many others who have already worked on this subject, but I am convinced that the truth will prevail.

The study of the infinite is much more important than one might think; it is the foundation of all true mathematics.

God created the integers; all else in mathematics is the work of man.

There is no one way to express the divine; mathematics is one of them.

The higher infinities are not beyond the reach of human thought.

My theory stands as firm as a rock, and I am convinced of its truth.

Infinity is a reality, not just a concept.

The continuum is something between the aleph-null and the next cardinal.

I have never proceeded from the implicit assumption of a correspondence between reality and our conceptual world.

The actual infinite is greater than any transfinite.