"Mathematical logic is the study of formal logic within mathematics."
Logic is the study of reasoning and argumentation. It includes topics like propositional logic, predicate logic, set theory, and proof theory.
Propositional Logic: Study of logical relationships between propositions involving logical connectives and truth values.
First-order Logic: Study of logical relationships between objects and variables using quantifiers and predicates.
Set Theory: Study of sets and functions, and their relationship.
Model Theory: Study of models of theories and their properties.
Proof Theory: Study of proofs and their properties.
Recursion Theory: Study of computability and undecidability.
Modal Logic: Study of propositions that express modalities, such as necessity and possibility.
Intuitionistic Logic: Study of constructive proofs and their properties.
Fuzzy Logic: Study of propositions that express degrees of truth, rather than strict true or false values.
Non-classical Logics: Study of logics that do not conform to classical logic, including paraconsistent and relevance logics.
"Major subareas include model theory, proof theory, set theory, and recursion theory."
"Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power."
"It can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics."
"Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics."
"This study began in the late 19th century."
"The development of axiomatic frameworks for geometry, arithmetic, and analysis."
"It was shaped by David Hilbert's program to prove the consistency of foundational theories."
"Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program."
"And clarified the issues involved in proving consistency."
"Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets."
"There are some theorems that cannot be proven in common axiom systems for set theory."
"Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems."
"Which parts of mathematics can be formalized in particular formal systems (as in reverse mathematics)."
"Trying to find theories in which all of mathematics can be developed."
"Mathematical logic is the study of formal logic within mathematics."
"Proof theory, set theory, recursion theory."
"Motivated by the study of foundations of mathematics."
"David Hilbert's program to prove the consistency of foundational theories."
"Characterize correct mathematical reasoning or to establish foundations of mathematics."