"Mathematical logic is the study of formal logic within mathematics."
The study of inference and deduction, as well as the theory of sets and functions, which provide the language for mathematical discourse.
Propositions and Truth Values: An introduction to the basic concepts of propositions and truth values.
Logical Connectives: An introduction to the different types of logical connectives.
Truth Tables: An introduction to truth tables and how they help in understanding the logical connectives.
Propositional Logic: An introduction to propositional logic, which deals with propositions and logical connectives.
First-Order Logic: An introduction to first-order logic, which is an extension of propositional logic and deals with predicates and quantifiers.
Deductive Systems: An introduction to the different types of deductive systems used in logic, including natural deduction and sequent calculus.
Formal Languages: An introduction to formal languages used in logic, including the syntax and semantics of such languages.
Predicate Logic: An introduction to predicate logic, which extends first-order logic and deals with the quantification of variables.
Set Theory: An introduction to the basic concepts of set theory, including sets, elements, subsets, set operations, and cardinality.
Axiomatic Set Theory: An introduction to axiomatic set theory, which develops set theory from a set of axioms.
Zermelo-Fraenkel Set Theory: An introduction to Zermelo-Fraenkel set theory, which is the most commonly used axiomatic set theory.
Axiom of Choice: An introduction to the axiom of choice, which is a controversial axiom used in set theory.
Topology: An introduction to topology, which studies the properties of space and continuity.
Model Theory: An introduction to model theory, which studies the properties of mathematical structures.
Proof Theory: An introduction to proof theory, which studies the properties of mathematical proofs.
Classical Logic: This is the traditional formal system used in mathematics, which is based on a binary approach of true/false or yes/no propositions.
Intuitionistic Logic: This type of logic is based on the idea that a proposition is only true if it can be proved to be so.
Modal Logic: This is a type of logic that formalizes the concepts of necessity and possibility, allowing for analysis of statements that involve these notions.
Fuzzy Logic: This is a type of logic used to deal with situations where the boundaries between true and false are not clearly defined or blurring.
Non-classical Logic: This is an umbrella term for various types of logic, which depart from classical logic in different ways—for example, by rejecting the law of non-contradiction, or by introducing more than two truth values.
First-order Logic: This is a formal system that uses quantifiers to describe the relationships between individuals and predicates in a language.
Second-order Logic: This is an extension of first-order logic that allows quantification over sets of objects.
Set Theory: This is the mathematical study of sets, which are collections of objects. Different forms of set theory include Zermelo-Fraenkel set theory, which is the dominant framework for the study of sets, and New Foundations Set Theory, which seeks to resolve some of the paradoxes and limitations of Zermelo-Fraenkel set theory.
Category Theory: This is a branch of mathematics that studies the mathematical properties of categories, which are generalized structures that unify different mathematical concepts.
Type Theory: This is a formal system in which objects are categorized according to their type, and which plays a crucial role in the foundations of mathematics, computer science, and linguistics.
"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."