# Intuitive Understanding of Order of Hyperreals

I'm trying to understand the ultrapower construction of the hyperreal numbers as described in Wikipedia.

The motivation is given as the surprising ability to find a total order of sequences of real numbers. Apparently if we select a free ultrafilter $$U$$ of the natural numbers, then it can be used to determine which indices "matter" in the set and then (from the article):

We write $$(a_0, a_1, a_2, ...) ≤ (b_0, b_1, b_2, ...)$$ if and only if the set of natural numbers $$\{ n : a_n ≤ b_n \}$$ is in $$U$$.

This is where I don't follow. Are we still working component-wise and then "$$\land$$-ing" the resultant sequence together as mentioned previously in the article?

The "intuitive explanation" which follows briefly mentions a parallel to Cantor's construction of the reals ... did I miss a step where this became a Cauchy sequence by implication?

In short: why does choosing a special set $$U$$ of indices of sequences to "matter" create a total order relation?

$$U$$ isn't a set of indices, it's a set of sets of indices. The idea is that two sequences are equal (or rather, name the same element) if they agree "most of the time" in the sense of $$U$$. Similarly, one sequence $$A$$ (names an element of the ultrapower which) is less than or equal to (the element named by) another sequence $$B$$ iff "most of the time"each entry of $$A$$ is $$\le$$ the corresponding entry of $$B$$. And so forth.

For example, consider the following sequences:

• $$A_1=(1,0,0,0,0,...)$$

• $$A_2=(0,1,0,0,0,...)$$

• $$A_n$$ has a $$0$$ in every place except the $$n$$th place, where it has a $$1$$.

I claim that these sequences are all equal (or rather, name the same element) in the ultrapower! For every $$i,j$$, the set of $$n$$ such that the $$n$$th elemenet of $$A_i$$ is $$=$$ the $$n$$th element of $$A_j$$ is cofinite, and every cofinite set is in $$U$$ since $$U$$ is a nonprincipal ultrafilter, so we always have $$[A_i]_U\le [A_j]_U$$ (this is a bit different from what you wrote - I'm being careful to distinguish between the sequence $$A_i$$ and the equivalence class $$[A_i]_U$$ which it belongs to; remember that elements of the ultrapower are equivalence classes of sequences, not individual sequences).

How does this generate a total order?

Well, suppose I have two sequences $$A=(a_n)_{n\in\mathbb{N}}$$ and $$B=(b_n)_{n\in\mathbb{N}}$$. Let $$X=\{n: a_n\le b_n\},\quad Y=\{n: a_n>b_n\}.$$ Clearly $$X\sqcup Y=\mathbb{N}$$, so since $$U$$ is an ultrafilter exactly one of $$X$$ and $$Y$$ is in $$U$$. If $$X\in U$$, then $$[A]_U\le [B]_U$$; if $$Y\in U$$, then $$[A]_U>[B]_U$$.

• Awesome thank you, I can see I should spend some more time with ultrafilters and ultrapowers! – KCE Feb 12 at 7:18
• I'm not sure how chat works, I thought I created a room that would invite you to it but it doesn't seem that way now. For a little more discussion on the topic I'd appreciate your attention to: chat.stackexchange.com/rooms/89602/… – KCE Feb 12 at 9:42