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Assignment 1, Math 712
1. Prove the Tarski-Vaught Lemma: Suppose that M ⊂ N are L-structures. Moreover, assume that whenever ϕ(y, x1, . . . , xn) is an L-formula such that N |= ∃yϕ(y, a1, . . . , an) for a1, . . . , an ∈ M then there is b ∈ M such that N |= ϕ(b, a1, . . . , an). Then the conclusion is that M ≺ N .
2. Prove the Lo´s Theorem.
3. Prove that if (Mi : i < α) is an elementary chain of L-structures i.e. Mi ≺ Mj whenever i < j < α, then the natural L-structure formed on the union of the Mi ’s is an elementary extension of each Mi .
4. Show that if (ri : i ∈ I) is a bounded I-indexed set of real numbers then limU ri is well-defined for any ultrafilter U on I.
5. Prove that if (X, d) is a compact metric space and U is any u.f. then Q U(X, d) ∼= (X, d).