## The infinite vs. the whole

So what’s the difference between the integers and the infinite integers? There is nothing in the second that is not in the first, so what in the world can “infinite” add?

When you speak of numerals as infinite you mean something like no point in the enumeration hits the last thing that can be enumerated. This assumes that “infinite” always begins with some part of the integers and denies something of it. If this is right, “infinite” is a judgment about a part or a way of considering something as never whole.

Briefly:

If infinite means “no part is the last one”, then infinite is a claim made about parts.

Though the infinite is never whole, that which is infinite can form a whole. This does not happen by adding something “on the end” or by filling out the rest of the process. There is no end to add things to, nor a “rest of the process” to go through. The whole of which “infinite integers” is a part is just “the integers”; the latter being a whole while the former is never is.