“Great innovation only happens when people aren’t afraid to do things differently.”

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

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

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My beautiful proof lies all in ruins.

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.

This view of the infinite, which I consider to be the sole correct one, is held by only a few. While possibly I am the very first in history to take this position so explicitly, with all of its logical consequences, I know for sure that I shall not be the last!

A set is a Many that allows itself to be thought of as a One.

Had Mittag-Leffler had his way, I should have to wait until the year 1984, which to me seemed too great a demand!

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.

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.

The essence of mathematics lies precisely in its freedom.

A false conclusion once arrived at and widely accepted is not easily dislodged and the less it is understood the more tenaciously it is held.

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.

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

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

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.

You should not be afraid of doing what your mind tells you to do, just listen to your heart.

Girls should never be afraid to be smart.

I am not the least afraid to die.

When we speak we are afraid our words will not be heard or welcomed. But when we are silent, we are still afraid. So it is better to speak.

Sometimes you’ve got to go to the wrong place just to show that you’re not afraid to go there.

Don’t be afraid of what you want. This is your time. The barriers are down.

Maybe there is something you’re afraid to say, or someone you’re afraid to love, or somewhere you’re afraid to go. It’s gonna hurt. It’s gonna hurt because it matters.

People are afraid of shadows. People are even more afraid of being in the shadow. But without a shadow, life will be two dimensional.