An example concerning Scott heights

Abstract

Unless otherwise stated, every structure in this paper is countable in a countable, actually recursive language and every formula is one of .The definition of the so-called canonical Scott-sentence of a structure M, CSS(M) (compare Nadel [8]), is based on the ordinal invariant called the Scott-height of M, denoted SH(M) (compare Makkai [5]). To describe SH(M), let “b ≡αα”, for finite tuples of equal lengths b and a of elements of M and any ordinal α, stand for “b and a satisfy (in M) the same formulas of quantifier-rank ≤ α” (for the quantifier rank of a formula, see e.g., Barwise [1]); also, let “b ∼ a” mean that there is an automorphism of M mapping b to a. With a fixed M, and a in M, sh(a) (or shM(a)) denotes the least ordinal α such that for all b in M, b ≡αa implies b ∼ a. SH(M) is the least ordinal α such that for all a and b in M, b ≡αa implies b ∼ a; hence SH(M) = sup{sh(a): a in M}.