Computable structure theory of continuous logic
Caleb M. H. Camrud · 2022
This dissertation examines computable structure theory relative to continuous logic. Due to the continuous nature of the relevant structures and space of truth values, special care is required to translate and modify results given in classical computable structure theory to the continuous setting. Three primary results are proven: (1) a generalized effective completeness theorem for continuous logic and computable presentations, (2) the existence of numerals for hyperarithmetical real numbers coded by computable infinitary sentences, and (3) upper and lower bounds on the complexity of the various diagram levels of the finitary and infinitary theories of a computably presented metric structure. Also given are basic model-theoretic results, an explicit formulation of the computable infinitary formulas of continuous logic, propositions concerning infinitary connectives, and a novel combinatorial result which allows for the encoding of a quantifier via a series inequality