Logic for Computer Science by Jean H. Gallier

Logic for Computer Science by Jean H. Gallier

Author:Jean H. Gallier [Jean H. Gallier]
Language: eng
Format: epub
Publisher: Dover Publications, Inc.
Published: 2015-07-14T16:00:00+00:00


(a) Assume that A and B are elementary equivalent. Show that for every formula E(x) with at most one free variable x, E(x) is satisfiable in A if and only if E(x) is satisfiable in B.

(b) Let X = <a1, ..., an> and Y = <b1, ..., bn> be two finite sequences of elements in A and B respectively. Assume that (A, X) and (B, Y) are elementary equivalent and that A is countably saturated.

Show that for any set Γ(x) of formulae with at most one free variable x over the expansion LY such that every finite subset of Γ(x) is satisfiable in (B, Y), Γ(x) is satisfiable in (A, X). (Note that the languages LX and LY are identical. Hence, we will refer to this language as LX.)

(c) Assume that A and B are elementary equivalent, with A countably saturated. Let Y = <b1, ..., bn, ... > be a countable sequence of elements in B.

Prove that there exists a countable sequence X = <a1, ..., an,... > of elements from A, such that (A, X) and (B, Y) are elementary equivalent.

Hint: Proceed in the following way: Define the sequence Xn = <a1, ..., an> by induction so that (A, Xn) and (B, Yn) are elementary equivalent (with Yn = <b1, ..., bn>) as follows: Let Γ(x) be the set of formulae over LYn satisfied by bn+1 in (B, Yn).

Show that Γ(x) is maximally consistent. Using 2(b), show that Γ(x) is satisfied by some an+1 in (A, Xn) and that it is the set of formulae satisfied by an+1 in (A, Xn) (recall that Γ(x) is maximally consistent). Use these properties to show that (A, Xn+1) and (B, Yn+1) are elementary equivalent.

(d) If (A, X) and (B, Y) are elementary equivalent as above, show that



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