netzweltler
Posts:
472
From:
Germany
Registered:
8/6/10


Re: It is a very bad idea and nothing less than stupid to define 1/3 = 0.333...
Posted:
Oct 8, 2017 5:40 PM


Am Sonntag, 8. Oktober 2017 22:12:22 UTC+2 schrieb Jim Burns: > On 10/8/2017 8:20 AM, netzweltler wrote: > > Am Sonntag, 8. Oktober 2017 13:23:05 UTC+2 > > schrieb FromTheRafters: > >> It happens that netzweltler formulated : > > >>> I'd have to look it up: Did you say that 0.999... IS the > >>> result of infinitely many addition operations or IS NOT > >>> the result of infinitely many addition operations? > >> > >> If you had an oracle with enough time which could do the > >> arithmetic and hand you an answer, then yes. Without such > >> an oracle, then I'd have to say no. That's why I said "after" > >> doing infinitely many steps you would have that number > >> exactly. John Conway used language similar to 'after infinitely > >> many of these steps, there is an explosion of sets...' to > >> describe a similar notion in describing his construction of > >> the surreals, so I'm not exactly breaking any new ground here. > > > I don't see the "infinite time" problem. What is the time a > > single addition operation takes? 0? More? Even if you don't > > allow 0 time for a single addition operation, try this: > > t = 0: Add 0 + 0.9 > > t = 0.9: Add 0.9 + 0.09 > > t = 0.99: Add 0.99 + 0.009 > > ... > > Every operation take some time greater than 0. Nonetheless, > > we have done infinitely many additions by t = 1. > > No "infinite time" involved. > > Not infinite time, but "enough time" for the oracle to operate, > as Mr Rafters said. It just happens that your oracle takes > 1 [unit of time] to operate. > > > No oracle needed. > > I don't see how you get this answer without an oracle. Maybe > you have an oracle and you don't realize it, it's invisible > or something. > > However, in some way you are satisfied that you can get an answer > for your infinite additions. Good. That was your criticism of > infinitely many additions, wasn't it? That there was no answer to > the _infinite_ sum? But now you _do_ have an answer. Somehow. > > (Sadly, I do not have an oracle, so I will continue to > define 0.999... the same way.) > >  > How does the oracle's answer (or whatever answer you apparently > have) compare to the standard answer? > > One thing we all know, including you, is that the sum of _all_ > of 0.9, 0.09, 0.009, ... must be strictly larger than any > _partial_ sum. So it can't be less than or equal to any of > 0.9, 0.99, 0.999, ...
This statement makes sense only if you can show that the sum of _all_ of 0.9, 0.09, 0.009, ... is a _point_on_the_number_line_.
> So, we know the oracle's answer is a bound of the set > { 0.9, 0.99, 0.999, ... }
I strongly disagree here. It cannot be a bound, because the numbers of that set represented as line segments in this list
[0, 0.9] [0, 0.9]?[0.9, 0.99] [0, 0.9]?[0.9, 0.99]?[0.99, 0.999] ...
already contain all the segments that you can find in 0.999... There is nothing to the right of all of those segments in 0.999...
> Suppose that we have two _distinct_ bounds b and c of all the > finite partial sums, with b < c. Whether or not b is the > oracle's answer to the infinite sum, we know that c _is not_ > the oracle's answer. > > Suppose c _was_ the infinite sum 0.9 + 0.09 + ... > Then we know what the "error term" in the infinite sum > is _exactly_ . > > For example, the sum of all the infinitely many terms after > the (10^100)th one > 9/10^(10^100+1) + 9/10^(10^100+2) + ... > is simply > (1/10^(100)*( 0.9 + 0.09 + 0.009 + ... ) = c/10^100 > > This gives us (by oracular means) a second way to calculate > the first 10^100 terms of the sequence: > 0.9 + 0.09 + ... + 9/10^100 > = (1  1/10^100)*c > > But this is a finite sum! > And b is also a bound of finite sums! > It had better be the case that > (1  1/10^100)*c < b < c > > But Wait! There's More! > Whatever nonzero distance b is below c, there is some > finite stage in the infinite sum 0.9 + 0.09 + ... > (if not 10^100, then 10^200, or 10^(10^100) or more) > where we can repeat this argument and contradict the > assumption that c is the _infinite_ sum. > > Therefore, if b and c are bounds and b < c, then c > _is not_ the oracle's answer. > > In fact, if the oracle answers with anything _other than_ > the least upper bound of finite partial sums, it will be > provably wrong. > >  > So, netzweltler, what answer did you come up with? Is it > different from 1?

