And while almost all real numbers aren't computable (by the argument in the article), essentially every number you'd ever stumble upon in a math class is.
Not every number though. It turns out that there are numbers that are describable but not computable.
What is the difference? Well here's an example. A Chaitin omega number is the probability that a valid program randomly constructed according to a specific set of rules will eventually halt. It is describable - indeed I just described it. Yet Chaitin has proven that an algorithm to compute it will lead to an algorithm that solves the halting problem, so it cannot be computable.
So there you have it. A class of perfectly describable numbers that are not computable. Indeed it can be proven that we cannot know more than a fixed number of digits for any particular one without increasing the size of our axiom system.
Not every number though. It turns out that there are numbers that are describable but not computable.
What is the difference? Well here's an example. A Chaitin omega number is the probability that a valid program randomly constructed according to a specific set of rules will eventually halt. It is describable - indeed I just described it. Yet Chaitin has proven that an algorithm to compute it will lead to an algorithm that solves the halting problem, so it cannot be computable.
So there you have it. A class of perfectly describable numbers that are not computable. Indeed it can be proven that we cannot know more than a fixed number of digits for any particular one without increasing the size of our axiom system.