Let (N, 0, S, +, *, E,...

 

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Let (N, 0, S, +, *, E,<,=) be the standard model of arithmetic (number theory). True or False? Explain why! 1. If is a countable structure that is elementary equivalent to then the two structures are isomorphic. 2. If AE then for every element a in the domain of 2 there are finitely many b such that b<a. 3. Let be a consistent set of formulas in the finite language L without equality. Let A be the structure with domain consisting of all L-terms, in which c=c for every constant c, f(t1,...tn) = ft... tn for every n-ary function symbol f and R (t₁...tn) if and only if Rt...tn. Then for every L-sentence σ we have 2 σ if and only if Σσ. 4. The class of all structures that are isomorphic to a fixed finite structure 2 in the language of arithmetic is elementary in the wider sense. 5. Let T be a consistent axiomatizable theory in the language of arithmetic. Then Th(n) T. 6. Η Σ Ε - and Σ Υ ψ then Σιφνψ. 7. Every theory with an infinite model has at least two non-elementary equivalent models. 8. Let f and g be representable in AE unary functions on the natural numbers. Then for every nЄN the function h N2 N defined by h(x, y) f(g(x))+g"(f(y)) is also representable in Ae (Recall fº(x) = x, ƒn+¹(x) = f(f(x))). = Let (N, 0, S, +, *, E,<,=) be the standard model of arithmetic (number theory). True or False? Explain why! 1. If is a countable structure that is elementary equivalent to then the two structures are isomorphic. 2. If AE then for every element a in the domain of 2 there are finitely many b such that b<a. 3. Let be a consistent set of formulas in the finite language L without equality. Let A be the structure with domain consisting of all L-terms, in which c=c for every constant c, f(t1,...tn) = ft... tn for every n-ary function symbol f and R (t₁...tn) if and only if Rt...tn. Then for every L-sentence σ we have 2 σ if and only if Σσ. 4. The class of all structures that are isomorphic to a fixed finite structure 2 in the language of arithmetic is elementary in the wider sense. 5. Let T be a consistent axiomatizable theory in the language of arithmetic. Then Th(n) T. 6. Η Σ Ε - and Σ Υ ψ then Σιφνψ. 7. Every theory with an infinite model has at least two non-elementary equivalent models. 8. Let f and g be representable in AE unary functions on the natural numbers. Then for every nЄN the function h N2 N defined by h(x, y) f(g(x))+g"(f(y)) is also representable in Ae (Recall fº(x) = x, ƒn+¹(x) = f(f(x))). =