Am Sonntag, 1. Oktober 2017 15:20:16 UTC+2 schrieb FromTheRafters: > After serious thinking netzweltler wrote : > > Am Sonntag, 1. Oktober 2017 13:56:01 UTC+2 schrieb FromTheRafters: > >> > >> It seems counterintuitive when a number is viewed (or represented) as > >> an infinite unending 'process' of achieving better and better > >> approximations, and that we can never actually reach the destination > >> number. In my view, this sequence and/or infinite sum is a > >> representation of the destination number "as if" we could have gotten > >> there by that process. > > If the process doesn't get us there then we don't get there. Where do you get > > your "as if" from? > > If you had sufficient time, then you would get there. Show how time is involved in our process.
> > >> IOW "*After* infinitely many 'better' > >> approximations" we reach the destination number *exactly* even if we > >> cannot 'pinpoint' that number on the number line. > > Please define "*After* infinitely many 'better' approximations". All we've > > got is infinitely many approximations - each approximation telling us that we > > get closer to 1 but don't reach 1. There is no *after* specified in this > > process. > > There is also no "time" mentioned, so why is there an assumption of a > process which takes time to complete? It is already completed (pi > exists as a number despite our inability to pinpoint it on the number > line by using an infinite alternating sum or any of the other infinite > processes) we just can't pinpoint it because we exist in a time > constrained universe with processes which take time to complete. If you insist on introducing time to our process, try this:
t = 0: write 0.9 t = 0.9: append another 9 t = 0.99: append another 9 ...
By time t = 1 we have completed infinitely many steps and we know all we need to know about our process: Since time is continuous we reach time t = 1 and after. By t = 1 we have completed writing 0.999... Since the steps of addition are discrete, we can tell that we don't reach point 1 - neither during the process nor *after* the process by t = 1.
If your claim is, that we reach point 1, you need to show which step on this _complete_ list of infinitely many steps accomplishes that.
> > > >> A 'limit' is not an > >> approximation, it is the destination number (if there is one in that > >> field) implied by the sequence or series in question. > > You could define a sequence or series by progressing from zero, to zero > plus one, to zero plus one plus one half, to zero plus one plus one > half plus one quarter, etcetera. This looks like it goes on forever > getting closer and closer to some number without actually ever getting > there. > > You could also define the same sequence or series by starting from two > and pulling something from one toward you by half the remaining > distance each time. In this second case, you already know the > destination even though the other representation of the same sequence > looks like it never gets there. Using the concept of infinity as > in,"infinitly many steps" you relieve yourself of the neccessity of > calculating the infinite sum approximations since you already know the > destination number (known as a limit). Despite the fact that the > process itself is neverending, these two 'things' are both > representations of the same number - namely two.