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 9, 2017 3:58 AM


Am Montag, 9. Oktober 2017 02:22:22 UTC+2 schrieb Jim Burns: > On 10/8/2017 5:40 PM, netzweltler wrote: > > 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_. > > This is _your definition_ . _My_ definition does not have > infinite sums in it. > > If this thing you're calling an infinite sum does not > yield a point on the number line, then it is not an infinite > sum in any useful sense. A sum yields a number. (And, if that > were so, I recommend using my ( _the_ ) definition for 0.999... ) > > If it turns out that this thing you're calling an infinite sum > _does_ yield a point, then  by the previous argument (which > I snipped for space)  the only point it can yield is the > least upper bound of the finite partial sums of the infinite > decimal  the same as my definition, only you use infinite sums > where I do not. (And, if that were so, it wouldn't matter which > you use, yours or mine.) > > Go ahead and take your time deciding whether your "infinite sum" > yields a point or not. Either way, 0.999... = 1. > > >> 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... > > I'm going to be charitable and assume that you don't know > what a bound is. The reason that 2 is a bound of the entries > in the sequence > 0.9, 0.99, 0.999, ... > is that > 0.9 =< 2 > 0.99 =< 2 > 0.999 =< 2 > ... > There is no requirement for 2 to be in any of your line segments > in order to be a bound. > > [...]
I'm going to be charitable and assume that you didn't get my argument. 1 and 2 is to the right of all of the segments ([11/10^(n1), 11/10^n])n?N, 0.999... is not.

