Georg Cantor Quotes (10 Quotes)

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 disimilar to, and I might even say in priciple the same as, my method described above of introducing trasfinite numbers. One can say unconditionally the transfinite numbers stand or fall with the finite irrational numbers they are like each other in their innermost being for the former like the latter are definite delimited forms or modifications of the actual infinite.

What I assert and believe to have demonstrated in this and earlier works is that following the finite there is a transfinite (which one could also call the supra-finite), that is an unbounded ascending lader of definite modes, which by their nature are not finite but infinite, but which just like the finite can be determined by well-defined and distinguishable numbers.

What a welcome relief this must have been to the chronically depressed Cantor As John D. Barrow writes in The Infinite Book A Short Guide to the Boundless, Timeless and Endless, ... started to tell his friends that he had not been the inventor of the ideas about infinity that he had published. He was merely a mouthpiece, inspired by God to communicate parts of the mind of God to everyone else.

The essence of mathematics lies in its freedom.

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

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

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.

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.

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

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