“The set of busy beaver numbers” communicates, points to, refers to, picks out, a concept in natural language that we can both talk about. It has finite information content (a finite sequence of ascii characters suffices to communicate it). An analogous sentence in a sufficiently formal language would still have finite information content.
Note that a description, even if it is finite, is not necessarily in the form that you might desire. Transforming the description “the sequence of the first ten natural numbers” into the format “{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}” is easy, but the analogous transformation of “the first 10 busy beaver numbers” is very difficult if not impossible.
As nshepperd points out, an element of a countable set necessarily has finite information content (you can pick it out by providing an integer—“the fifth one” or whatever), while generic elements of uncountable sets cannot be picked out with finite information.
“The set of busy beaver numbers” communicates, points to, refers to, picks out, a concept in natural language that we can both talk about. It has finite information content (a finite sequence of ascii characters suffices to communicate it). An analogous sentence in a sufficiently formal language would still have finite information content.
Note that a description, even if it is finite, is not necessarily in the form that you might desire. Transforming the description “the sequence of the first ten natural numbers” into the format “{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}” is easy, but the analogous transformation of “the first 10 busy beaver numbers” is very difficult if not impossible.
As nshepperd points out, an element of a countable set necessarily has finite information content (you can pick it out by providing an integer—“the fifth one” or whatever), while generic elements of uncountable sets cannot be picked out with finite information.