You really do not like the Axiom of Choice 
The Axiom of Choice implies that non-measurable sets exist.
I’m not big with numbers (heh), but I just happened to watch a really great documentary on Netflix called: A trip to Infinity.
The feeling I got from the documentary is that infinity is a lot like Schrodinger’s Cat. Infinity is all numbers until you observe it.
(Loved the infinity hotel paradox)
Isn’t that a fundamental trick in set theory? One starts with a universe, and expands it to a larger universe having the property you want. So you change things like “all natural numbers”. AIUI this is how to prove the independence of the continuum hypothesis.
Yes, this does seem like an extension just like complex numbers from the reals. Every ‘real’ number could be considered the sum of a real and ‘imaginary’ part where the latter is zero. So maybe we can add an extra ‘dimension’ of infinitesimals which also are not part of the ‘real number line’ but work ‘outside it’? I guess tht was roughly what Robinson was thinking?
The closest analogy to the complex numbers would be the dual numbers cited above, where you insert an epsilon such that \epsilon^2=0. That leads to a two-dimensional number system, similar to the complex numbers. It’s sufficient for calculus purposes (I think).
Robinson though worked with a full-fledged hyperreal system, with not just a single infinitesimal value, but an infinite number of ever-smaller ones, as well as an infinite tower of infinities (which are just 1 \over \epsilon, 1 \over \epsilon^2, etc.). So it’s a bit more general than the dual numbers.
I wonder if there’s any utility to “triple numbers”, “quad numbers”, etc., where \epsilon^N=0, but \epsilon^{N-1}>0, and \epsilon>\epsilon^2>\epsilon^3, .... The universe seems to only require some finite number of derivatives for its operation. Perhaps it’s enough to only store a few layers of infinitesimals.
You have put your finger on it precisely. The pieces that appear in B-T are non-measurable. The existence of such sets requires AC.
Yes, precisely. I’d rather take measurability of sets as an axiom than take the Axiom of Choice.
What; no love for the generalized continuum hypothesis? 
Aleph-1 all the way!