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 ≡ α α ”, 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 sh M ( a )) denotes the least ordinal α such that for all b in M , b ≡ α a implies b ∼ a . SH( M ) is the least ordinal α such that for all a and b in M , b ≡ α a implies b ∼ a; hence SH( M ) = sup{sh( a ): a in M }.