The transfinite numbers are in a sense the new irrationalities [ ... they] stand or fall with the finite irrational numbers.

Don't always blindly follow guidance and step-by-step instructions; you might run into something interesting.

To ask the right question is harder than to answer it.

There is no doubt that we cannot do without variable quantities in the sense of the potential infinite. But from this very fact the necessity of the actual infinite can be demonstrated.

Mathematics, in the development of its ideas, has only to take account of the immanent reality of its concepts and has absolutely no obligation to examine their transient reality.

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.

My beautiful proof lies all in ruins.