#### Title of Research Output

Contemporary Mathematical Approaches to Computability Theory

#### Faculty

Science

#### Supervisor Name

Rasul Shafikov

#### Keywords

computability theory, turing machines, undecidability, computable numbers, computable functions, computable real functions

#### Description

In this paper, I present an introduction to computability theory and adopt contemporary mathematical definitions of computable numbers and computable functions to prove important theorems in computability theory. I start by exploring the history of computability theory, as well as Turing Machines, undecidability, partial recursive functions, computable numbers, and computable real functions. I then prove important theorems in computability theory, such that the computable numbers form a field and that the computable real functions are continuous.

#### Acknowledgements

I would like to thank Professor Rasul Shafikov for his exceptional guidance and feedback during the USRI and my writing of this paper.

#### Document Type

Paper

#### Included in

Analysis Commons, Number Theory Commons, Other Mathematics Commons, Theory and Algorithms Commons

Contemporary Mathematical Approaches to Computability Theory

In this paper, I present an introduction to computability theory and adopt contemporary mathematical definitions of computable numbers and computable functions to prove important theorems in computability theory. I start by exploring the history of computability theory, as well as Turing Machines, undecidability, partial recursive functions, computable numbers, and computable real functions. I then prove important theorems in computability theory, such that the computable numbers form a field and that the computable real functions are continuous.