Am Mittwoch, 4. Oktober 2017 13:43:34 UTC+2 schrieb FromTheRafters: > netzweltler was thinking very hard : > > Am Mittwoch, 4. Oktober 2017 11:44:35 UTC+2 schrieb Zelos Malum: > >>> In fact it means exactly infinitely many commands. > >>> But of course if you define a series to be equal to its limit, then that's > >>> like defining an apple equal to an orange. That is your problem, not ours. > >> > >> It doesn't because it is not operations upon operations, it is just a > >> representation of one element in real numbers. > > > > It does. And 0.875 is representing 3 operations, e.g. > > 0.8 + 0.07 + 0.005 or > > 0.5 + 0.25 + 0.125. > > > > It can be an element AND represent some number of operations. > > I agree. The thing is that a finite number of steps (or commands) can, > at best, give a good enough approximation of some numbers.
The 3-step operations above give the number exactly. You can get this number after any finite number of operations. Whereas you can't get the number 0.874999... after a finite number of operations.
> By adding > 'ad infinitum' to the 'end' of these, and assuming it can be completed > in that (NaN) number of steps, you can arrive at the exact answer. This > is a case where it is not about the trip, but about the (eventual) > destination being defined exactly. > > As you already know, an arrow traveling from zero toward a target at > two which can be described as going halfway there then halfway the > remaining distance, then halfway again, may seem like an unending > process with only better and better approximations being attainable. > But, if I can get that same step by step process by stating that the > process is actually pulling the arrow half the distance (from zero to > two for example) then I don't need the unending process anymore as I > already have the destination and the fact that the process is the same > confirms that two is the answer I want. > > Insisting on the process while standing two units in front of the > archer will not save you from getting the point. :)