Arvutatavad

Arvutatavad, also known as "computable numbers" in English, are a fundamental concept in the theory of computation and mathematical logic. They were first introduced by the mathematician Alan Turing in his 1936 paper "On Computable Numbers, with an Application to the Entscheidungsproblem." The concept is central to the field of computability theory, which studies the fundamental capabilities and limitations of computers.

A number is considered arvutatav if it can be computed by an algorithm. In other words, a number is arvutatav if there exists a step-by-step procedure that, given enough time and resources, will produce the number as an output. This definition is closely related to the concept of a Turing machine, an abstract model of computation that can simulate the logic of any computer algorithm, no matter how complex.

The set of arvutatavad numbers is countably infinite, meaning that it can be put into a one-to-one correspondence with the natural numbers. This is significant because it shows that there are only countably many numbers that can be computed by an algorithm, even though the set of all real numbers is uncountably infinite. This implies that there are real numbers that are not arvutatavad, and thus cannot be computed by any algorithm.

The study of arvutatavad numbers has important implications for the philosophy of mathematics and the foundations of computer science. It provides a rigorous framework for understanding what it means for a number to be computable and helps to delineate the boundaries of what can and cannot be computed by an algorithm.