From Cantor's Attic
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The Greeks had already noted that there are two ways of considering infinity.

  • Potential infinity is what we consider when we say that counting never ends. Whatever natural number you can think of, there is a bigger number. Formally

$$(\forall x\in\mathbb{N})(\exists y) (y>x),$$ and this is not really deniable.

  • Actual infinity is what happens when we switch the order: There is a number which is bigger than any natural number you can think of. Formally

$$(\exists y)(\forall x\in\mathbb{N})(y>x),$$ and this naturally implies that $y$ cannot be a member of $\mathbb{N}$.

The existence of an actual infinity is philosophically non trivial and not accepted by all mathematicians. Its existence cannot be proven; it is axiomatically given (see ZFC).

The real question however is, given a collection, how does one determine whether it is finite or not. Certainly, if one can count all the elements of a collection, then the collection is finite, but who can count up to a googol? yet it is finite.

Gallileo noticed that there are as many even numbers as there are positive integers. To see this without formal machinery: imagine the collection of all positive integers. Don't try to imagine them individually, imagine them as a completed collection. Now multiply them all by 2. There are no left overs. This violates the Greek saying that the whole is greater than the parts.

This leads to Dedekind's characterisation: A finite set cannot be in one-to-one relation with a proper subset. An infinite set is a set that can be in one-to-one relation with a proper subset.

In this sense, once accepted the existence of the set of all natural numbers, $\mathbb{N}$ is infinite since it is possible to map $n$ to $n+1$ thus providing a one-to-one relation from $\mathbb{N}$ to $\mathbb{N}\setminus\{0\}$.