For years I’m contemplating about a countable subset of R, bigger than Q:
The set of numbers which can be determined with a finite number of characters.
This set (I call it R- or sub-transcendental numbers) is obviously countable.

What are the properties of R-?
What are the properties of R \ R- (I call it R+): A number of R+ cannot be described with a finite number of characters (in no way). You cannot grasp a number in R+ (if you could, it would be in R-)

See here the beginning of a paper (not too serious) in which I try to formalise my thoughts.

More is coming up.