An Introduction to Computability Theory

Abstract

What are the capabilities and limitations of computers? Are they glorified adding machines capable of superfast arithmetic computations and nothing else? Can they outdo man in the variety of problems they can handle? Let’s narrow the question a bit. Consider the class of number theoretic functions that a computer can be programmed to compute or that a man can be instructed to compute. Are any of these functions computable by a computer but not by a man, or by a man but not by a computer? Is there a number theoretic function that is not computable by any computer, and if so, can such functions be described? Is there a computer that can be programmed to compute any function that any other computer can compute? Is man such a computer?