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

(A thesis defended at Cantor's doctoral examination)

The essence of mathematics lies in its freedom.

Quoted in D MacHale, *Comic Sections * (Dublin 1993)

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

*Je le vois, mais je ne le crois pas!*

I see it but I don't believe it!

Letter to R Dedekind regarding results of *Theory of Manifolds* (1877)

My theory stands as firm as a rock; every arrow directed against it will quickly return to the 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.

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

I realise 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.

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

The essence of mathematics lies in its freedom.