Enderton 2.2.25

Consider a fixed structure $\displaystyle A$. Expand the language by adding a new constant symbol $\displaystyle c_a$ for each $\displaystyle a \in |A|$. Let $\displaystyle A^+$ be the structure for this expanded language that agrees with $\displaystyle A$ on the original parameters and that assigns to $\displaystyle c_a$ the point $\displaystyle a$. A relation $\displaystyle R$ on $\displaystyle |A|$ is said to be definable from points in $\displaystyle A$ iff $\displaystyle R$ is definable in $\displaystyle A^+$. (This differs from ordinary definability only in that we now have parameters in the language for members of $\displaystyle |A|$.) Let $\displaystyle K=(\mathbb{Re}; <, +, \cdot)$.

(a) Show that if $\displaystyle X$ is a subset of $\displaystyle \mathbb{Re}$ consisting of the union of finitely many intervals, then $\displaystyle X$ is definable points in $\displaystyle K$.

(b) Assume that $\displaystyle A \equiv K$. Show that any subset of $\displaystyle |A|$ that is non-empty, bounded (in the ordering $\displaystyle <^A$), and definable from points in $\displaystyle A$ has a least upper bound in $\displaystyle |A|$.

(*)$\displaystyle A \equiv K$ iff for any sentence $\displaystyle \sigma$, $\displaystyle \models_A \sigma \Leftrightarrow \models_K\sigma$.

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To get started,

Why $\displaystyle c_a$ and $\displaystyle A^+$ are introduced in this problem?

How an open set and a closed set are defined by formulas in $\displaystyle A^+$?

Thanks.