Even if you do have infinitesimals, no reasonable definition of 0.99 repeating is anything except 1.0: Suppose you have a number space with a) a sequence of a

Even if you do have infinitesimals, no reasonable definition of 0.99 repeating is anything except 1.0: Suppose you have a number space with a) a sequence of approximations 0.9, 0.99, ... (not necessarily indexed by the naturals, just some infinite set that gets bigger) b) 1.0 c) enough "niceties" like (a-b)!=0 for a!=b, addition existing, yada yada Suppose that some "solution" exists in that number space for your favorite way of writing 0.99 repeating. One reasonable thing to expect from such a solution is that your sequence of approximations actually get closer to (or eventually equal) it. That one constraint (that your better approximations are actually better approximations) forces any possible solution to be exactly equal to 1.0. That doesn't necessarily mean the solution is 1.0, just that if it isn't (and is something infinitesimally different) then one of the assumptions must have been violated. Unfortunately, most of the assumptions were simply describing the problem, so the only real possibilities for failure (if we assume some solution exists) are the foundations of math being fundamentally flawed (it might happen, but that doesn't seem plausible here), or the number space we're considering being considerably less "nice" than the reals (i.e., not nice enough for most of modern physics and whatnot). You could of course instead say that it's dumb that we can define 0.99 recurring to actually equal something, but that's just quibbling over the names and representations that we're' giving to things -- sequences of better approximations exist, 1.0 exists, and there exists some partial function turning sequences of approximations into their "limits", and calling this particular instance something other than 0.99 recurring doesn't really get rid of the fundamental idea. All of that aside, you can of course get infinitesimal approximations in other spaces (like the hyperreals), and those infinitesimal approximations are better than any real approximations, but the _actual definition_ of 0.99 repeating isn't equal to any of those approximations -- they just progressively get closer to 1.0.