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Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory , proof theory , set theory , and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power.
FOL is now a core formalism of mathematical logic, and is presupposed by contemporary treatments of Peano arithmetic and nearly all treatments of axiomatic set theory. The 1928 edition included a clear statement of the Entscheidungsproblem ( decision problem ) for FOL, and also asked whether that logic was complete (i.e., whether all semantic ...
These problems were also studied by mathematicians, and this led to establish mathematical logic as a new area of mathematics, consisting of providing mathematical definitions to logics (sets of inference rules), mathematical and logical theories, theorems, and proofs, and of using mathematical methods to prove theorems about these concepts.
The term "mathematical logic" is sometimes used as a synonym of "formal logic". But in a more restricted sense, it refers to the study of logic within mathematics. Major subareas include model theory, proof theory, set theory, and computability theory. [164] Research in mathematical logic commonly addresses the mathematical properties of formal ...
This is a list of mathematical logic topics. For traditional syllogistic logic, see the list of topics in logic . See also the list of computability and complexity topics for more theory of algorithms .
Pure inductive logic (PIL) is the area of mathematical logic concerned with the philosophical and mathematical foundations of probabilistic inductive reasoning. It combines classical predicate logic and probability theory ( Bayesian inference ).
Equational logic was common before Principia Mathematica (e.g. Johnson (1892)), and has present-day advocates (Gries & Schneider (1993)). Conventional mathematical logic consists of tautological formulae, signalled by a prefixed turnstile. To denote that the primary algebra formula A is a tautology, simply write "A =".
Logic is the foundation that underlies mathematical logic and the rest of mathematics. It tries to formalize valid reasoning. In particular, it attempts to define what constitutes a proof. List of Boolean algebra topics; List of first-order theories; List of large cardinal properties; List of mathematical logic topics; List of set theory topics