Abstract

Model Theory. These are non-empty families I of partial isomorphisms between models M and N , closed under taking restrictions to smaller domains, and satisfying the usual Back-and-Forth properties for extension with objects on either side -- restricted to apply only to partial isomorphisms of size at most k . 'Invariance for k--partial isomorphism' means having the same truth value at tuples of objects in any two models that are connected by a partial isomorphism in such a set. The precise sense of this is spelt out in the following proof. 21 Proof (Outline.) k-variable formulas are preserved under partial isomorphism, by a simple induction. More precisely, one proves, for any assignment A and any partial isomorphism IÎI which is defined on the A-values for all variables x 1 , ..., x k , that M, A |= f iff N , IoA |= f . The crucial step in the induction is the quantifier case. Quantified variables are irrelevant to the assignment, so that the relevant partial isomorphism can be res...

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