Item — HN Reader

Hyperreal numbers make Leibniz notation literal and algebraically consistent and rigorous, rather than a convenient shorthand for limits.

Author suggests using them. Resistance to Hyperreals seems to come from era before there were rigorous definitions for them.

1 comments

Personally I dislike hyperreals for the reason that we lose the all-powerful Archimidean property (for all real numbers x, there exists a natural number N such that N > x or N = x). But yes, hyperreals are completely consistent. Robinson showed, I want to say in the 1960s, that any statement about the logical consistency of the hyperreals is true if and only if the same is true for the reals. Then hyperreals basically save Leibniz by introducing the standard part function st(), which is rigorously defined.

Again, for me, this seems a little costly. If cookbook calculus is a struggle, the student probably won't take real analysis. And once you get to analysis, the fact that the reals are Archimidean grounds the whole endeavor on a very intuitive basis. By contrast, the hyperreals are not Archimedian so we can find omega bigger than every natural number. I don't personally feel like that helps intuition and indeed requires development of non-standard analysis but I admit it is pretty cool that Leibniz-style computations can be salvaged this way.

I think the cost is just a question of tradition what you find harder and the point author makes that it's easier to learn trough non-traditional way.

I see losing Archimedean property as more of a win than loss. It's a one simple thing that turns calculus operations into single algebraic evaluation, where you can avoid dynamic limits. Things become simpler and easier to prove.