Georg Cantor Quotes

Some infinities are bigger than other infinities.

We must say that there are as many squares as there are numbers.

No one shall expel us from the paradise that Cantor has created for us.

There are infinite numbers between 0 and 1.

The Axiom of Choice is necessary to select a set from an infinite number of socks, but not an infinite number of shoes.

A googolplex is precisely as far from infinity as is the number 1... no matter what number you have in mind, infinity is larger still.

[On Cantor's work:] The finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.

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.

To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed.

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.

Infinites, when considered absolutely without any restriction or limitation, are neither equal nor unequal, nor have any certain proportion one to another, and therefore, the principle that all infinites are equal is a precarious one.

Set theory in sheep's clothing.

I entertain no doubts as to the truths of the tranfinites, which I recognized with God's help and which, in their diversity, I have studied for more than twenty years; every year, and almost every day brings me further in this science.

Mathematics is concerned only with the enumeration and comparison of relations.

It is almost as hard to define mathematics as it is to define economics, and one is tempted to fall back on the famous old definition attributed to Jacob Viner, "Economics is what economists do," and say that mathematics is what mathematicians do. A large part of mathematics deals with the formal relations of quantities or numbers.

The old and oft-repeated proposition "Totum est majus sua parte" [the whole is larger than the part] may be applied without proof only in the case of entities that are based upon whole and part; then and only then is it an undeniable consequence of the concepts "totum" and "pars". Unfortunately, however, this "axiom" is used innumerably often without any basis and in neglect of the necessary distinction between "reality" and "quantity", on the one hand, and "number" and "set", on the other, precisely in the sense in which it is generally false.

For mathematics, even to the logical forms in which it moves, is entirely dependent on the concept of natural number.

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.

An increasing number of people are growing uncomfortable with the gulf between the world's rich and the poor. Ostentatiously splashing your money around simply draws attention to that disparity, and to your own position on the lucky higher ground. It suggests a callousness, an inhumanity, a let's-just-rub-their-noses-in-it arrogance.

The theory of numbers, more than any other branch of mathematics, began by being an experimental science. Its most famous theorems have all been conjectured, sometimes a hundred years or more before they were proved; and they have been suggested by the evidence of a mass of computations.

We all start out so damn sure, thinking we've got the world on a string. If we ever stopped to think about the infinite number of ways we could be undone, we'd never leave our bedrooms.

If the poor, for example, because they are more in number, divide among themselves the property of the rich,- is not this unjust? . . this law of confiscation clearly cannot be just.

I hope to make acting my career for the rest of my life, if I can. If acting doesn't work out, I'd love to produce or direct or write. I just want to stay in this business, definitely. That would be my number one thing. I always want to be an actress.

I have the absolute confidence not to be number two, but then I have enough sense to realize that there can be no number one.

When a large number of organs of the press come to advance along the same track, their influence becomes almost irresistible in the long term, and public opinion, struck always from the same side, ends by yielding under their blows.