>>14014014I do not understand the big issue with this construction of the real numbers, but the most reasonable ones are objections to the use of something called the axiom of choice and the assumption that you can even have an infinite set. The beauty of math is that if you disagree with any assumptions, go right ahead and refuse to assume them. There are many things being done in mathematical logic and often they involve altering your assumptions.
To answer your question of whether 99.9999...=100 or not, all we need to do is unpack our definitions, make some observations, then repack them. 99.9999... is just the set of all sequences whose difference with 90, 99, 99.9, 99.99, ... converges to 0. 100 is the set of all sequences whose difference with 100, 100, 100, 100, ... converges to 0. If we look at the differences between these two sequences, we get
10, 1, 0.1, 0.01, 0.001, ...
By the definition of convergence, for an arbitrarily small number x, we can find some n such that the nth number in the sequence 10, 1, 0.1, 0.01, ... is less than x. Therefore, this sequence converges to 0. Therefore, we conclude with absolute certainty from our definitions that 99.9999...=100.
Like I said, you have every right to explore what happens when you assume different things, and that is not at all uncommon in mathematics. But if you don't want to assume the same things that I have, there is no point trying to argue about them. Something can only be true up to the assumptions you are making, and under these assumptions, 99.9999...=100 is true.
