firstname.lastname@example.org has brought this to us : > On Saturday, September 30, 2017 at 2:42:46 PM UTC-7, netzweltler wrote: >> Am Samstag, 30. September 2017 23:14:36 UTC+2 schrieb mitchr...@gmail.com: >>> >>> .9 repeating and One share a sameness. They are quantities >>> that are different by the infinitely small. >>> .9 repeating is a transcendental One; the First quantity >>> below one. The infinitely small difference means a shared >>> sameness that is still not absolutely same. >>> >>> Mitchell Raemsch >> >> If there is a quantity between 0.999... and 1 > > At some point there needs to be next quantities with > nothing in between.
Not really, in fact in the reals if there are two different (nearly?) adjacent real numbers then there is always a place between them for another real number to occupy.
Even worse would be the case of two nearly adjacent rationals. If 0.999 repeating (a rational number) were to be taken as one of those numbers and 1.000 repeating (another rational number) as the other then it is easy to see that in the reals, many numbers (irrationals) would have to exist between these two. The fact remains that these two numbers are actually only two representations of the same exact number.
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. IOW "*After* infinitely many 'better' approximations" we reach the destination number *exactly* even if we cannot 'pinpoint' that number on the number line. 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.
> and, therefore, these are two different points on the number line then you > should define the distance between these two points. If you don't, then your > first quantity is simply undefined.
I know that you are against the idea that a number can have multiple representations (like 0.999... and 1.000...) but it happens all the time. 1/1 2/2 3/3 etcetera all represent the number one.
>> 'infinitely small' is not a definition. > > It has a definition of being one divided by infinity > It can't be divided any further. It is The Infinitely divided One. > Their quantity difference is by the infinitely small. > This means there are no quantities in between them. > >> There are no two distinct points on the number line 'infinite(simal)ly' far >> away from each other. > > > Mitchell Raemsch
Thank you for defending your view rather than just restating your view. It makes for a much better discussion.