Enderton 2.2.25

Consider a fixed structure . Expand the language by adding a new constant symbol for each . Let be the structure for this expanded language that agrees with on the original parameters and that assigns to the point . A relation on is said to be definable from points in iff is definable in . (This differs from ordinary definability only in that we now have parameters in the language for members of .) Let .

(a) Show that if is a subset of consisting of the union of finitely many intervals, then is definable points in .

(b) Assume that . Show that any subset of that is non-empty, bounded (in the ordering ), and definable from points in has a least upper bound in .

(*) iff for any sentence , .

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To get started,

Why and are introduced in this problem?

How an open set and a closed set are defined by formulas in ?

Thanks.