The calculus does deal with infinities but doesn’t place them in a rigorous framework.
I’m not sure that the 19th century knew what Archimedes had done. The Archimedes palimpsest wasn’t found until 1906 and a critical passage in it not until 2001.
Yes, but what does it prove? To me it just says if you could write an infinite string of numbers (which of course you can’t do), you could write an infinite string of 9s preceded by “0.”
Is it true? Either way it doesn’t have that much to do with math(s) to me as maths is about abstract notions.
The limit clearly is 1. It is the lowest value which is higher than any element of the sequence (0.9, 0.99, 0.999, …). I still don’t think that this is hard to understand.
Did you quote the right post?
But we don’t need a rigorous framework for infinities. We only need a rigorous framework for limits. Which is exactly what calculus is.
For the record, the limit is clearly 1, but the reasoning (‘lowest value which is higher than any element of the sequence’) isn’t really how limits are defined. You could have a sequence [such as (1/n)*(-1)^n, for n=1, 2, 3, 4,…] that has a limit, but converges on the limit from both above and below.
The question to solve is whether “reaching the limit” is one of the actions given on this list or not:
t = 0: I move my pen from point 0 to point 0.9 of the number line
t = 0.9: I move my pen from point 0.99 to point 0.999 of the number line
t = 0.99: I move my pen from point 0.999 to point 0.9999 of the number line
…
Which point of the number line my pen is pointing to at t >= 1?
You have not defined what happens when t is greater than 1. Which is why definitions are so important.
Or do you mean when t = 1? Which is why agreed upon symbolism is so important.
On the list you do not find definitions for t = 1 and t > 1 either. All I want to know is which point the pen is pointing to after completing the actions given on the list!
ONE.
It is pointing at 1.
IT’S ONE.
The actions on the list don’t allow me to reach point 1. Each of the infinitely many lines of the list shows how I get closer to 1 but not reach 1. So, what does make you thinking it’s pointing to 1?
Even if you tell me 1000 times that it’s 1: The list tells me infinitely many times that it’s not 1.
The list tells you where it’s pointing when t<1. With a bit of expansion you can show that the limit is 1 and the limit is 0.999… so 0.999… = 1.
You don’t understand what the infinite list is saying.
(My corrections)
As far as I can tell, your pen is moving with constant speed. When t is 1 your pen is at 1.
Why don’t you just tell us what you want us to say?
Apparently, netzweltler is confused by thinking that Zeno’s paradox still is unsolved. The poor arrow never reaches the tree because it’s confused by thinking it’s in some infinitely long series of actions. 
Yes, with a bit of expansion. But the bit of expansion is not on the list.
O yes. It says, don’t reach point 1.
The pen is not moving at constant speed - not as I have defined it. It is moving in discrete steps. There is no action defined for t = 1 on the list.
What I want to make clear is, that moving the distance 0.999… does not get us to a defined point of the number line - not to point 1 and not to a point short of 1 either.
Zeno’s dichotomy paradox has never been solved.