Georg Cantor

The Absolute can only be acknowledged and admitted, never known, not even approximately known.

The distinction between the 'transfinitum' and the 'absolutum' seems to me to be fundamental and indispensable.

A point-set P is called 'perfect' if it coincides with its derivative P'.

By an 'aggregate' or 'set' I understand any collection into a whole M of definite, distinct objects m of our intuition or our thought.

Two sets M and N are equivalent... if it is possible to set them in a relation such that to every element of each one corresponds one and only one element of the other.

The power of a set M will be denoted by |M|.

The system of all numbers is an inconsistent, absolutely infinite multiplicity.

The continuum problem is the question of what power the linear continuum has.

I have never proceeded from any 'Genus supremum' of the actual infinite. Quite the contrary, I have rigorously proven that there is absolutely no 'Genus supremum' of the actual infinite. What surpasses all that is finite and transfinite is no 'Genus'; it is the single, completely individual unity in which everything is included, which includes the 'Absolute', incomprehensible to the human understanding. This is the 'Actus Purissimus' which by many is called 'God'.

The transfinite numbers are not less real than the finite irrational numbers.

My dear friend, your reproach that I have not sufficiently studied Kronecker's works is unjust; I know them, perhaps better than you, and have found nothing in them that could shake my theories.

The so-called 'ultrafinite' numbers of my opponents are nothing but my transfinite numbers in disguise.

I am convinced that the time will come when my ideas will be recognized as simple, true, and indispensable for the foundation of analysis.

The diagonal proof shows that the continuum cannot be mapped one-to-one onto the set of natural numbers; thus, there are different sizes of infinity.

The highest perfection of a mathematical theory is reached when it attains such a degree of simplicity and generality that one can say: from now on, this is self-evident.

Poetry is as exact as geometry.

The true mathematician is a passionate artist.

I realize that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers.

The transfinite sequence of numbers is, in a certain sense, a new irrationality. It is, as it were, the irrationality of the second order.

The concept of the continuum presented here is, I believe, the only correct one.