Georg Cantor

The theory of sets is a powerful tool for abstract thinking, which allows us to reason about complex systems.

The transfinite numbers are a testament to the power of pure reason to transcend the limitations of our senses.

The theory of sets is a fundamental part of logic, which provides a rigorous framework for mathematical reasoning.

The transfinite numbers are a challenge to our preconceived notions about the nature of numbers and infinity.

The theory of sets is a universal language for all of mathematics, which allows us to unify different branches of mathematics.

The transfinite numbers are a testament to the power of abstraction to reveal the hidden structures of reality.

The theory of sets is a powerful tool for exploring the foundations of mathematics and for developing new mathematical theories.

The transfinite numbers are a source of endless fascination and wonder for all who delve into their mysteries.

The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number, or order type.

Mathematics is entirely free in its development, and its concepts are only linked by the necessity of being consistent, and they are not dictated by the external world of phenomena.

The fear of infinity is a form of myopia that destroys the possibility of seeing the actual infinite, even though it in its highest form has created and sustains us, and in its secondary transfinite forms occurs all around us and even inhabits our minds.

Every transfinite consistent multiplicity, i.e., every transfinite set, must have a definite aleph as its cardinal number.

My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all, because I have followed its roots, so to speak, to the first infallible cause of all created things.

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, delineated forms or modifications of the actual infinite.

A false conclusion once arrived at is not easily set aside, and the more ingenious the person who drew the conclusion, the more difficult it is to rid oneself of it.

The potential infinite means nothing other than an undetermined, variable quantity, always remaining finite, which has to assume values that either become smaller than any finite limit no matter how small, or greater than any finite limit no matter how great.

By a 'set' we mean any collection M into a whole of definite, distinct objects m (which are called the 'elements' of M) of our perception or of our thought.

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 diagonal proof is so simple that a child of ten can understand it.

The collection of all alephs is not a set, but an inconsistent multiplicity.