## 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.

1. #### Will Farris said,

January 28, 2016 at 2:30 pm

This is the whole basis of integral calculus, which is a process of summation of an infinite number of terms. A convergent series such as those generated by the Zeno paradoxes (1 + 1/2 + 1/4 + 1/8 +…) eventually adds up, or converges, to 2. Thus the number 2 is the totality of a certain “real” infinity, even though we cannot actually put physical objects in any one to one correspondence to the terms.

• #### James Chastek said,

January 28, 2016 at 2:35 pm

Converging on something requires not reaching it. One doesn’t add up to 2 but treats it as a term of pure becoming. If you wanted to just get to 2, there are much easier ways than by taking a limit (you can just advance by units, for example, and get there in two steps.)