Index Sets of Computable Structures

The index set of a computable structure A is the set of indices for computable copies of A. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary finite structures, Q-vector spaces, Archimedean real closed ordered fields, reduced Abelian p-groups of length less than ω, and models of the original Ehrenfeucht theory. The index sets for these structures all turn out to be m-complete Π n , d-Σ n , or Σ n , for various n. In each case, the calculation involves finding an “optimal” sentence (i.e., one of simplest form) that describes the structure. The form of the sentence (computable Πn, d-Σn, or Σn) yields a bound on the complexity of the index set. When we show m-completeness of the index set, we know that the sentence is optimal. For some structures, the first sentence that comes to mind is not optimal, and another sentence of simpler form is shown to serve the purpose. For some of the groups, this involves Ramsey theory. ∗Calvert acknowledges the support of the NSF grants DMS-0139626 and DMS-0353748, Harizanov of the NSF grant DMS-0502499, and Miller of the NSF grant DMS-0353748. Harizanov also gratefully acknowledges the support of the Columbian Research Fellowship of the George Washington University.