Defining the Infinite

“There is no smallest among the small and no largest among the large;
But always something still smaller and something still larger.” (Anaxagoras)

“A quantity is infinite if it is such that we can always take a part outside what has been already taken.” (Aristotle, Physics)

“For generally the infinite has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite […].” (Aristotle, Physics)

“What is that thing which does not give itself and which if it were to give itself would not exist? It is the infinite!” (Leonardo da Vinci)

“When we say anything is infinite, we signify only that we are not able to conceive the ends and bounds of the thing named.” (Thomas Hobbes)

“We call infinite that thing whose limits we have not perceived, and so by that word we do not signify what we understand about a thing, but rather what we do not understand.” (René Descartes)

“I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a facon de parler [manner of speaking], the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without restriction.” (Carl F Gauss, 1831)

“An infinite set is one that can be put into a one-to-one correspondence with a proper subset of itself.” (Georg Cantor)

“The Infinite is often confounded with the Indefinite, but the two conceptions are diametrically opposed. Instead of being a quantity with unassigned yet assignable limits, the Infinite is not a quantity at all, since it neither admits of augmentation nor diminution, having no assignable limits; it is the operation of continuously withdrawing any limits that may have been assigned: the endless addition of new quantities to the old: the flux of continuity. The Infinite is no more a quantity than Zero is a quantity. If Zero is the sign of a vanished quantity, the Infinite is a sign of that continuity of Existence which has been ideally divided into discrete parts in the affixing of limits.” (George H Lewes, “Problems of Life and Mind”, Vol. 2, 1875)

“A collection of terms is infinite when it contains as parts other collections which have just as many terms in it as it has. If you can take away some of the terms of a collection, without diminishing the number of terms, then there is an infinite number of terms in the collection.” (Bertrand Russell. International Monthly, Vol. 4, 1901)