5
votes
3answers
632 views

Prove that transcendental numbers exist: Are there less paniful ways of doing it?

I've found this exercise on Boolos' Logic and Computability: A real number $x$ is called algebraic if it is a solution to some equation of the form: $$c_{\small d}x^{\small ...
3
votes
2answers
181 views

Are some numbers more computable than others?

As I understand it (layman alert), the definition of computable numbers is binary: either a number is or is not computable. Is it meaningful to imagine a function telling how computable (or ...
4
votes
3answers
255 views

Numbers which are “Provably Difficult to Compute”?

We recall that a computable number $\alpha \in \mathbb{R}$ satisfies the following: there exists a computable function $f$ such that, given any positive rational error bound, $f$ outputs a rational ...
10
votes
1answer
180 views

Is there a dense subset of $\mathbb{R}^2$ with all distances being incommensurable?

Is there a set $S$ of points on the real plane $\mathbb{R}^2$ such that: there is a point belonging to $S$ in any neighborhood of every point of $\mathbb{R}^2$ (so, $S$ is dense) and ratio of any ...