Computable structure theory of continuous logic
Date
2022-05
Authors
Camrud, Caleb Matthew
Major Professor
Advisor
McNicholl, Timothy
Biggs, Stephen
Herzog, David
Lutz, Jack
McCullough, Jason
Weber, Eric
Committee Member
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Abstract
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
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dissertation