Individual km#mathematical_logic__symbolic_logic__metamathematics__metamathematic
  >part of:  km#formal_logic  km#abstract_algebra
  url:  http://en.wikipedia.org/wiki/Mathematical_logic
  >part:  km#application_of_techniques_of_formal_logic_to_mathematics
     >part:  km#logicism  pioneered by philosopher-logicians such as Gottlob Frege and Bertrand Russell: the idea was that mathematical theories were logical tautologies, and the programme was to show this by means to a reduction of mathematics to logic. The various attempts to carry this out met with a series of failures, from the crippling of Frege's project in his Grundgesetze by Russell's Paradox, to the defeat of Hilbert's Program by Gödel's incompleteness theorems; however, every rigorously defined mathematical theory can be exactly captured by a first-order logical theory; Frege's proof calculus is enough to describe the whole of mathematics, though not equivalent to it
  >part:  km#application_of_mathematical_techniques_to_formal_logic
  >part:  km#sentential_logic__propositional_logic__propositional_calculus  proof theory for reasoning with propositional formulas as symbolic logic; it is extensional
  >part:  km#predicate_logic__predicatelogic__predicate_calculus  permits the formulation of quantified statements such as "there is at least one X such that..." or "for any X, it is the case that...", where X is an element of a set called the domain of discourse
     >part:  km#FOL__first-order_logic__firstorderlogic__first-order_predicate_calculus__predicate_logic__predicatelogic  FOL is distinguished from HOL in that it does not allow statements such as "for every property, it is the case that..." or "there exists a set of objects such that..."; it is a stronger theory than sentential logic, but a weaker theory than arithmetic, set theory, or second-order logic; it is strong enough to formalize all of set theory and thereby virtually all of mathematics
        >part:  km#PCEF_logic__logic_of_positive_conjunctive_existential_formulas
     >part:  km#HOL__higher-order_logic  based on a hierarchy of types
  >part:  km#logic_for_reasoning_about_computer_programs
     >part:  km#Hoare_logic
     >part:  km#logic_for_reasoning_about_concurrent_processes_or_mobile_processes
        >part:  km#CSP
        >part:  km#CCS
        >part:  km#pi-calculus
     >part:  km#logic_for_capturing_computability__logicforcapturingcomputability
        >part:  km#computability_logic__computabilitylogic

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