Relativize Pseudo-finite Homogeneity and Isolation

Theorem 1

Let U be a universe. Let an extended r-query Q be locally generic for finite states over U.

Suppose, for some uncountable k with k=k^*, there is a special model (V,J) of power k such that J is an indiscernible sequence in V, V is first-order equivalent to U, and the query Q is locally generic over pseudo-finite states over J in V. Then Q is equivalent over finite states over U to a restricted r-query.

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