Mathematical Logic

Church-Turing Thesis: Claim that every computable function can be computed by a Turing machine. Computability theory: Study of computable functions on the natural numbers. Continuum hypothesis: Conjecture that there are only two sizes of infinite sets of real numbers. Database: Finite, typically relational structure. First order logic: Mathematical model of the part of language built up from the propositional connectives and the quantifiers. Incompleteness phenomenon: Gödel’s discovery, that sufficiently strong axiomatic theories cannot decide all propositions which they can express. Model theory: Study of formal definability in first order structures. Paradox: Counterintuitive truth. Peano arithmetic: Axiomatic theory of natural numbers. Proof theory: Study of inference in formal systems independently of their interpretation. Propositional connectives: The linguistic constructs “and”, “not”, “or” and “implies”. Quantifiers: The linguistic constructs “there exists” and “for all”. Turing machine: Mathematical model of computing device with unbounded memory. Unsolvable problem: A problem whose solution requires a non-existent algorithm. Narrowly construed, mathematical logic is the study of definition and inference in mathematical models of fragments of language, especially the first order logic fragment. Logic has made critical contributions to the foundations of science, especially through the work of Kurt Gödel, and it also has numerous applications. For set theory and theoretical computer science, these applications are so important, that parts of these fields are normally included in the modern, broad conception of the discipline.