Mathematical logic (also recognized as symbolic logic) is a subfield of maths with close connections to the foundations of mathematics, theoretical computer science, and philosophical logic. The field incorporates both the mathematical examination of the philosophy and the applications of formal logic to other mathematics areas. The unifying themes in mathematical logic consist of the research of the emotional strength of formal methods and the deductive strength of formal evidence techniques.

Mathematical logic is usually divided into the fields of set concept, product principle, recursion principle, and proof theory. These places reveal final theoretical results on philosophy, notably very first-buy logic, and definability. In pc science (specifically in the ACM Classification), mathematical logic encompasses further subjects not comprehensive in this article; see a thesis in computer science for these.

It is considering that its inception, mathematical logic has contributed to the research of foundations of arithmetic. It started in the late 19th century with the growth of axiomatic frameworks for geometry, arithmetic, and analysis. In the early twentieth century, it was shaped by David Hilbert's plan to confirm the consistency of foundational concepts.

Outcomes of Kurt Godel, Gerhard Gentzen, and other individuals offered a partial resolution to the plan and clarified the troubles involved in proving consistency. Work in established ideas showed that practically all everyday arithmetic could be formalized in conditions of sets, even though some theorems are unable to be confirmed in standard axiom methods for the established concept. Modern-day perform in the foundations of arithmetic usually centers on establishing which components of arithmetic can be formalized in distinct formal programs (as in reverse mathematics) instead of attempting to uncover notions in which all of the arithmetic can be designed.

- Complex Examination

- Game Theory

- Computing

- Probability

- Combination

- Logics

- Tensor Examination

- Differential Calculus

- Analytic Geometry