Questions about mathematical logic, including model theory, proof theory, computability theory (a.k.a. recursion theory), and non-standard logics. Questions which merely seek to apply logical or formal reasoning to other areas of mathematics should not use this tag. Consider using one of the following tags as well, if they fit the question: (model-theory), (set-theory), (computability), and (proof-theory). This tag is not for logical puzzles, use (puzzle).
This tag broadly covers the field of mathematical logic, which deals with questions involving formalized mathematical statements, mathematical structures, and their relationships. The development of mathematical logic in the late 19th and early 20th centuries was intertwined with the interest in foundations of mathematics (foundations), although much current work in logic is not directly related to foundations.
The elementary content of mathematical logic involves formal mathematical languages, quantifiers, and formal proofs of statements. These formal proofs are carried out in formal proof systems (see formal-proofs), which model ordinary mathematical reasoning but, unlike natural language proofs, have a fully specified syntax and grammar that could in principle be verified mechanically. Specific tags for these topics include quantifiers and first-order-logic. The full development of these ideas happens in the field of proof-theory . A well known application of proof-theoretic methods is Gödel's incompleteness theorem incompleteness.
The field of model-theory studies models of formal languages. Examples include algebraic structures such as groups and rings, as well as more esoteric structures. The field focuses on definability within such structures, relative to particular formal languages.
The field of computability studies formalized notations of computability, such as Turing computability and hyperarithmetical computability, as well as their applications to mathematics.
The field of set-theory studies sets by considering formal axiomatic systems of set theory such as ZFC. Questions about basic topics that might be found in "Chapter 0" of an undergraduate textbook (such as unions, intersections, subsets, etc.) are classified on this site as elementary-set-theory , while the set-theory includes questions about models of ZFC, large cardinals, the method of forcing, etc. Some researchers view set theory as part of mathematical logic, while others view it as a distinct area; the logic tag is not mandatory for set theory questions.
There are other areas which overlap with mathematical logic, but are not always considered part of it. The field of category-theory has many similarities to logic, and has important foundational aspects.
The foundational aspects of logic include mathematical constructivism, which is classified here as constructive-mathematics.
This tag does not include questions about ordinary logical reasoning in mathematical proof writing. Questions that ask about the logical structure or logical methods of ordinary mathematical proofs should be labeled with the proof-writing tag unless they ask about specific formal proof systems.
This tag should not be used for what a layperson might called "a logical puzzle". For these sort of questions please use recreational-mathematics and puzzle as appropriate. (Unless the solution is done via a method relevant to the logic tag, of course.)