What Does It Mean For An Irrational Number To Be Computable? A computable number is a number that can be calculated by a finite computer program. All the numbers you have ever heard of like 3, √2, π, e, etc. are computable. Some numbers (like π) are represented by an infinite string of nonrepeating digits.
Are Real Numbers Uncomputable? Most real numbers can never be calculated, they’re uncomputable, which suggests that mathematics is full of things that we can’t know, that we can’t calculate. This is related to something famous called Gödel’s incompleteness theorem from 1931, five years before Turing. What makes a number computable? A computable number is a
Is ATM Recognizable? Because we know that ATM is recognizable, our theorem implies that ATM and ATM are both decidable. But we know that ATM is not decidable. This is a contradiction, hence ATM cannot be recognizable. The language ATM and its undecidability Is ATM co Turing? We know that ATM is Turing recognizable, but
Are All Functions Computable? Are all functions computable? I’d like to share a simple proof I’ve discovered recently of a surprising fact: there is a universal algorithm, capable of computing any given function! What functions are not computable? The set of finitary functions on the natural numbers is uncountable so most are not computable. Concrete