# Mathematical Logics Assignment Help

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**Mathematical logic** (also known as **symbolic logic**) is a subfield of **mathematics** with close connections to the **foundations** of **mathematics**, **theoretical** **computer** **science** and **philosophical** **logic**.[1] The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics. The unifying themes in mathematical logic include the study of the expressive power of **formal** **systems** and the deductive power of formal **proof** systems.

Mathematical logic is often divided into the fields of **set** **theory**, **model** **theory**, **recursion** **theory**, and **proof** **theory**. These areas share basic results on logic, particularly **first**-**order** **logic**, and **definability**. In computer science (particularly in the **ACM** **Classification**) mathematical logic encompasses additional topics not detailed in this article; see **logic** **in** **computer** **science** for those.

Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of **foundations** of **mathematics**. This study began in the late 19th century with the development of **axiomatic** frameworks for **geometry**, **arithmetic**, and **analysis**. In the early 20th century 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, although 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 (as in **reverse** **mathematics**) rather than trying to find theories in which all of mathematics can be developed.

**Types of Mathematical Logics:**

- Set theory and paradoxes
- Symbolic logic
- Formal logical systems
- Model theory
- Recursion theory
- Proof theory and constructive mathematics
- Connections with computer science
- Foundations of mathematics

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