Georg Cantor

Two sets are said to have the same power if they can be placed into a one-to-one correspondence with each other.

The continuum hypothesis is a very plausible proposition, whose proof, however, has so far not been found, despite intense effort.

The Absolute Infinite is the conception of the maximum, the perfect being, which many call God.

The finite is annulled in the infinite and becomes zero, but the infinite in the finite is not annulled and does not become zero.

The transfinite numbers are not less determinate than the finite numbers, but they are of a higher order of determinateness.

The set of all algebraic numbers is countable.

The power of the set of points on a line is greater than the power of the set of natural numbers.

The introduction of new numbers is justified by the fact that they serve to express relations between magnitudes that could not be expressed by the old numbers.

The infinite is not a quantity that can be increased or decreased; it is an absolute maximum.

The essence of the mathematical mind is not logic but imagination.

The set of all subsets of a set has a greater power than the set itself.

The transfinite numbers are the numbers of the infinite.

The concept of order type is as important for the theory of order as the concept of cardinal number is for the theory of size.

The truth of my theory will, in time, establish itself, for it is true.

The transfinite numbers are in a certain sense themselves new irrationalities and in fact in my opinion the best method of defining the finite irrational numbers is wholly dissimilar to, and I might even say in principle the same as, my method described above of introducing transfinite numbers. One can say unconditionally: the transfinite numbers stand or fall with the finite irrational numbers; they are alike in their most intrinsic nature; for the former like the latter are definite delimited forms or modifications of the actual infinite.

Had I been influenced by the sterile and hollow admiration of the multitude of mathematicians, or had I believed that my theory should be judged by its utility, I would have long since kept silent.

Mathematics is entirely free in its development, and its concepts are only linked by the necessity of being consistent, and are in no way constrained by the external world of phenomena.

The transfinite with its richness of forms and formations necessarily points towards an Absolute, towards the 'true infinite', whose magnitude is not subject to any increase or decrease and which is therefore to be regarded quantitatively as an absolute maximum.

The potential infinite is nothing but an auxiliary, relational concept, which only points to an unlimited coming-into-being, but never to a fixed, self-contained infinite quantity.

The finite has been the sole object of mathematics since ancient times. I was the first to investigate the infinite in a systematic way, and to recognize it in its various forms and magnitudes.