Earlier quoted context omitted.
The point isn’t to be “right” — whatever you define that as. The point is to be understood. Use blackboard bold when it’s what people are most familiar with and will engender the least confusion. You call this the “tabs vs spaces” debate, but I’ve literally never heard anyone advocate your position until you did, just now.
> I’ve literally never heard anyone advocate your position until you did, just now Many mathematicians concerned by typography use real boldface. For example: Terence Tao's blog, Donald Knuth, Paul Halmos (author of "how to write mathematics"), and the famous journal "Publications Mathématiques de l'IHÉS" which is the undisputed gold standard in mathematical typography. They use real boldface for the number sets N, Z…
BB is common, in my experience, in hand-written text where bold isn't really an option. It is, to my tradition, the most common and recognizable way to indicate the most common field sets and also generally is a good stand-in for any "large category" of interest.
Bold is used intermittently in my experience, probably due to its inability to be hand-written. To me, it tends to mean "vector" or "matrix" much more than set. In hand-written forms, I sometimes see "arrow hats" instead, especially for vectors.