I don't know if now it's just a confirmation bias but basically there is a question of numerability and there seem to be a way that this is elementarily built from R
"However, a 2003 paper by Vladimir Kanovei and Saharon Shelah[4] shows that there is a definable, countably saturated (meaning ?-saturated, but not, of course, countable) elementary extension of the reals, which therefore has a good claim to the title of the hyperreal numbers. Furthermore, the field obtained by the ultrapower construction from the space of all real sequences, is unique up to isomorphism if one assumes the continuum hypothesis."
also, I am very stupid because when I thought
2^aliph0^aliph0 I didn't realize that
aliph0^aliph0= aliph0 so the cardinality =_= is the same, I thought that aliph0^aliph0 >aliph0
for some reason.
So yeah, it's not bigger but it is countably saturated
for some reason I was thinking about a "basic function" while a function is a subset of cartesian product, so the most "basic" structure is extinguishing intersections of nonempty sets, infact all functions are intersections of the cartesian product imho but not all intersections are functions.
So, even if the cardinality is the same the countability is saturated in that sense. By "expanding" R to hyperreals you still have the cardinality R.
I tried to read the paper but it's way above my league, I'll just leave it to the wiki reference
Sorry for bad English, it's pretty late desu
