In Henson's Model Theory lecture notes I found an exercise quite early on (1.30, p. 12) that prove too difficult for me. It goes like this:
Let $L$ be the first order language whose only nonlogical symbol is the binary predicate symbol $<$. Let $\mathcal{A}=(\mathbb{N},<)$ and let $\mathcal{B}=\mathcal{A}^{I} / U$ be an ultrapower of $\mathcal{A}$ where $I$ is countably infinite and $U$ is a nonprincipal ultrafilter on $I$.
- Show that $\mathcal{B}$ is a linear ordering.
- Show that the range of the diagonal embedding of $\mathcal{A}$ into $\mathcal{B}$ is a proper initial segment of $\mathcal{B}$. Give an explicit description of an element of $B$ that is not in the range of this embedding.
- Show that $\mathcal{N}$ is not a well ordering; that is, describe an infinite descending sequence in $\mathcal{N}$.
For the first point, I realize I "just" need to check whether it is reflexive, transitive, antisymmetric and strongly connected. But I do not see how to even treat the definition of $\mathcal{B}$. Then I am completely lost at the second dot.
I would appreciate any help!
The first part is just an easy application of Łoś's theorem but is trivially proved directly.
The second part. If $a=((n))$ is an element of the diagonal image of $A$ and $b=((b_i))<a$, then almost for all $i$ $b_i<n$. Then there are only finitely many different $b_i$. Since the ultrafilter is not principal, there is a set $S$ from the ultrafilter and $s<n$ such that $b_i=s$ for all $i\in S$. Then $b=((s))$ belongs to the image of the diagonal embedding. QED
The third part is straightforward: just find an infinite descendent sequence starting with $(1,2,2^2, 2^{2^2}, 2^{2^{2^2}},...)$ (again using the fact that the ultrafilter is not principal).