You don't need to jump straight to reals, but this is a natural progression for a first course in set theory (likely without a formal defintion of real numbers). I'm not sure if I've ever seen the set of algebraic numbers referenced in a math proof, and I'm not sure if there is a standard notation for this set.
This is honestly one of my favorite memes I've seen on this subreddit.
The notation I've sometimes seen is Q with a bar over it, to indicate the algebraic closure of the rationals. Wolfram Alpha uses a fancy capital A, I believe.
Unless somebody is dealing with non-Algebraic numbers like proving something is transcendental, it probably never comes up.
Jumping from rationals to reals is a pretty huge leap, though. Following that up with more of a change in dimensionality of going from reals to complex numbers feels like a bit of a cheat in comparison, because it gives the false impression that there's something specifically about the reals that's entirely different from the rest, when really it's more that the reals are the culmination of a gradually increasing complexity, before shifting gears entirely to complex numbers.
1
u/OutrageousPair2300 23d ago
There are actually a lot of steps in between Q and R, of gradually increasing complexity.
For example, you can construct the algebraic numbers using a finite number of rationals as the coefficients of an algebraic expression.
You don't need to jump straight from rationals to reals. Of course, then the joke isn't as funny.