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.
Mathematics may be defined as the economy of counting. There is no problem in the whole of mathematics which cannot be solved by direct counting.
We must say that there are as many squares as there are numbers.
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.
No one shall expel us from the paradise that Cantor has created for us.
[On Cantor's work:] The finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.
And what is the greatest number? Number one.
Set theory in sheep's clothing.
The infinite in mathematics is alway unruly unless it is properly treated.
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.
Number is different from quantity.
