It baffles me how there are still blogposts with a serious attitude about this topic. It’s akin to discussing possible loopholes of how homeopathy might be medicinally helpful beyond placebo, again and again.
Why are hyperreals even mentioned? This post is not about hyperreals or non-standard math, it’s about standard math, very basic one at that, and then comes along with »well under these circumstances the statement is correct« – well no, absolutely not, these aren’t the circumstances the question was posed under.
We don’t see posts saying »1+2 = 1 because well acktchually if we think modulo 2«, what’s with this 0.9… thing then?
I think it's worse than this. Even with hyperreals, 0.999... = 1, I believe, since they have to obey all laws of arithmetic that are true for the reals. At the very least, 3 × 0.333... = 1, and not 0.999... even for the hyperreals.
IMHO the confusion arises, because the author failed to recognise that N cannot be a natural number if they go down the nonstandard analysis path. N would have to be elevated to a hyperinteger as well, which would eliminate the infinitesimal they end up with.
I'm saying that, in the hyperreals as well as the reals, I am 100% certain that 3 × 0.33... = 1. I am not as sure that 0.999 = 1 with the hyperreals, BUT, if it's true as the author claims that 0.99... ≠ 1 in the hyperreals, then it must follow that 3 × 0.33... ≠ 0.99... in the hyperreals.
I still think that the distinction is very important. With standard math (e.g. real numbers) we obviously have 0.9999... = 1 and this is actually very easy to prove using the assumptions that you are taught during high school math.
However, in higher math you are taught that all this is just based on certain assumptions and it is even possible to let go of these assumptions and replace them with different assumptions.
I think it is important to be clear about the assumptions one is making, and it is also important to have a common set of standard assumptions. Like high school math, which has its standard assumptions. But it is just as possible to make different assumptions and still be correct.
This kind of thinking has very important applications. We are all taught the angle sum in a triangle is 180 degrees. But again this is assuming (default assumption) euclidean geometry. And while this is sensible, because it makes things easy in day to day life, we find that euclidean geometry almost never applies in real life, it is just a good approximation. The surface of the earth, which requires a lot of geometry only follows this assumption approximately, and even space doesn't (theory of relativity). If we would have never challenged this assumption, then we would have never gotten to the point where we could have GPS.
It is easy to assume that someone is wrong, because they got a different result. But it is much harder to put yourself into someones shoes and figure out if their result is really wrong (i.e. it may contradict their own assumption or be non-sequitur) or if they are just using different assumptions. And to figure out what these assumptions are and what they entail.
For this assumption: Yes, you can construct systems where 0.9999... != 1, but then you also must use 1/3 != 0.33333... or you will end up contradicting yourself. In fact when you assume 1 = 0.999999... + eps, then you must likely also use 1/3 = 0.33333 - eps/3 to avoid contradicting yourself (I haven't proven the resulting axiom system is free of contradiction, this is left as an excercise to the reader).
Why are hyperreals even mentioned? This post is not about hyperreals or non-standard math, it’s about standard math, very basic one at that, and then comes along with »well under these circumstances the statement is correct« – well no, absolutely not, these aren’t the circumstances the question was posed under.
We don’t see posts saying »1+2 = 1 because well acktchually if we think modulo 2«, what’s with this 0.9… thing then?