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Topic: Re: It is a very bad idea and nothing less than stupid to define 1/3
= 0.333...

Replies: 42   Last Post: Oct 9, 2017 11:53 AM

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 netzweltler Posts: 473 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
> >> 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.)
>
> 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 ([1-1/10^(n-1), 1-1/10^n])n?N, 0.999... is not.

Date Subject Author
10/2/17 Guest
10/2/17 netzweltler
10/2/17 Jim Burns
10/3/17 netzweltler
10/3/17 FromTheRafters
10/3/17 Jim Burns
10/3/17 FromTheRafters
10/3/17 Jim Burns
10/3/17 FromTheRafters
10/3/17 netzweltler
10/3/17 bursejan@gmail.com
10/4/17 netzweltler
10/3/17 FromTheRafters
10/3/17 Jim Burns
10/3/17 FromTheRafters
10/3/17 netzweltler
10/3/17 Jim Burns
10/4/17 netzweltler
10/4/17 Jim Burns
10/4/17 netzweltler
10/5/17 Jim Burns
10/5/17 netzweltler
10/5/17 Jim Burns
10/5/17 netzweltler
10/5/17 Jim Burns
10/5/17 netzweltler
10/5/17 Jim Burns
10/5/17 FromTheRafters
10/6/17 netzweltler
10/6/17 Jim Burns
10/7/17 FromTheRafters
10/8/17 FromTheRafters
10/8/17 netzweltler
10/8/17 Jim Burns
10/8/17 netzweltler
10/8/17 Jim Burns
10/9/17 netzweltler
10/9/17 Jim Burns
10/9/17 netzweltler
10/9/17 Jim Burns
10/7/17 Jim Burns