A Programming Language Oriented Approach to Computability

The field of computability and complexity was, where computer science sprung from. Turing, Church, and Kleene all developed formalisms that demonstrated what they held "intuitively computable". The times change however and today's (aspiring) computer scientists are less proficient in building automata or composing functions and are much more native to the world of programming languages. This article will try to introduce typical concepts of computability theory and complexity in a form more fitted for a modern developer. It is mostly based on \cite{jones}, but takes input from other sources to provide examples, additional information, etc.