Showing that there exists some elementarily equivalent structure with a descending chain? This is question number 7 from Enderton's "A Mathematical Introduction to Logic":
Consider a language with a two-place predicate symbol $<$, and let $\mathfrak{R} = (\mathbb{N};<)$ be the structure consisting of the natural numbers with their usual ordering. Show that there is some $\mathfrak{A}$ elementarily equivalent to $\mathfrak{R}$ such that $<^{\mathfrak{A}}$ has a descending chain.
I am really stumped by this. Could anyone offer a hint?
 A: HINT: Are you familiar with the compactness theorem? Can you think of a way to describe an infinite descending chain - possibly by adding a (or several) constant symbol(s) to the language, and then writing some sentences about it/them?

Why this should be plausible: $\mathfrak{R}$ has descending sequences of arbitrary finite length - e.g. a descending sequence of length $17$ would be $17>16>15>...$. Now Compactness formally states: if every finite subset of $\Gamma$ is satisfiable, then $\Gamma$ is satisfiable; but a more intuitive formulation would be, "If every finite subset of $\Gamma$ 'occurs' in the structure $\mathfrak{R}$, then there is some elementary extension of $\mathfrak{R}$ where $\Gamma$ 'occurs'" (look at $Th(\mathfrak{R})\cup\Gamma$)). In this case, we're taking $\Gamma$ to be some set of sentences which, together, mean "There is an infinite descending sequence." So our job is to:


*

*Come up with the relevant $\Gamma$ (in a possibly expanded language - this is a common trick),

*Show that $\Gamma\cup Th(\mathfrak{R})$ is finitely consistent (using the fact that $\mathfrak{R}$ has arbitrarily long finite descending chains), and

*Show that any model of $\Gamma\cup Th(\mathfrak{R})$ (or rather, the reduct of any such model to the original language $\{<\}$) is an elementary extension of $\mathfrak{R}$ with an infinite descending chain.
This same outline works for a number of problems. It is one way to show, e.g., that $Th(\mathbb{N}; +, \times, <)$ has a proper elementary extension.
