“The essence of mathematics lies precisely in its freedom.”

In mathematics, the art of asking questions is more valuable than solving problems.

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

Mathematics is entirely free in its development, and its concepts are only linked by the necessity of being consistent, and are co-ordinated with concepts introduced previously by means of precise definitions.

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.

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.

Since my return to America my Master, has at the desire of my friends in England given me my freedom.

Because of them I can now live the dream. I am the seed of the free, and I know it. I intend to bear great fruit.

We shouldn’t be fearful of freedom, we shouldn’t be fearful of individual liberty.

In every human Beast, God has implanted a Principle, which we call Love of Freedom; it is impatient of Oppression, and pants for Deliverance.

To be Christians under the law of grace does not mean to wander unbridled outside the law, but to be engrafted in Christ, by whose grace we are free from the curse of the law, and by whose Spirit we have the law engraved upon our hearts.

He [Christ] died for me. He made His righteousness mine and made my sin His own; and if He made my sin His own, then I do not have it, and I am free.

Every person shall be free to do good at his own expense.

The word ‘free’ is used three times in the Declaration of Independence and once in the First Amendment to the Constitution, along with ‘freedom.’ The word ‘fair’ is not used in either of our founding documents.