Infinite Sets Quotes

Some infinities are bigger than other infinities.

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

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

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.

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.

Something is infinite if, taking it quantity by quantity, we can always take something outside.

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.

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.

Two infinities: the one that stretches to the beginning but never touches-when you halve and halve and halve, infinitely-and then the one that spreads out into the endless, endless future, the endless, endless, distance.The set of infinities that is itself infinite.

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.