[LiKao]

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).