Definibility in normal theories

This paper initiates an investigation which seeks to explain elementary definability as the classical results of mathematicallogic (the completeness, compactness and Löwenheim-Skolem theorems) explain elementary logical consequence. The theorems of Beth and Svenonius are basic in this approach and introduce automorphism groups as a means of studying these problems. It is shown that for a complete theoryT, the definability relation of Beth (or Svenonius) yields an upper semi-lattice whose elements (concepts) are interdefinable formulas ofT (formulas having equal automorphism groups in all models ofT). It is shown that there are countable modelsA ofT such that two formulae are distinct (not interdefinable) inT if and only if they are distinct (have different automorphism groups) inA. The notion of a concepth being normal in a theoryT is introduced. Here the upper semi-lattice of all concepts which defineh is proved to be a finite lattice—anti-isomorphic to the lattice of subgroups of the corresponding automorphism group. Connections with the Galois theory of fields are discussed.