Ok. I see. It is still not clear what I said. When I say that the position of 0.999… is not defined on the number line, then this means you cannot do any defined comparison or calculation. So, I never said that 0.999… < 1.
To take the square root of an undefined point on the number line will result in an undefined point again.
Didn’t I define that when I said there is a line for each n ∈ ℕ?
We don’t pick a particular line, we take all of them and see that none of them does get us to point 1.
(Copy and paste again)
n = 1, t = 0: I move my pen from point 0 to point 0.9 of the number line
n = 2, t = 0.9: I move my pen from point 0.9 to point 0.99 of the number line
n = 3, t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line
…
For every n ∈ ℕ there is a line on the list.
For every n ∈ ℕ is valid that we don’t reach 1.
If 0.999… would be the same point as point 1 on the number line we would have reached point 1 by t = 1.
Try that with -1:
n = 1, t = 0: I move my pen from point 0 to point -0.5 of the number line
n = 2, t = 0.9: I move my pen from point -0.5 to point -0.75 of the number line
n = 3, t = 0.99: I move my pen from point -0.75 to point -1 of the number line
n = 4, t = 0.999: I move my pen from point -1 to point -1 of the number line
n = 5, t = 0.9999: I move my pen from point -1 to point -1 of the number line
…
For n ≥ 3 is valid that we reached -1.
At t = 1 we can state that the pen is pointing to -1. So, -1 is a well-defined point on the number line.
At t = 1 the pen has jumped an infinite number of times. Nowhere the pen jumped to point 1.0. Otherwise, show where.
You haven’t shown it doesn’t … this is your case to make.
I did. Do you want me to copy and paste again?
Not on the number line under discussion.
God no! You’ve posted the same thing several dozen times in this thread and it hasn’t convinced anyone, and you’ve handwaved all the refutations. You’re out on an island here and that’s just fine. If you think you have something useful please demonstrate something useful you can do with your assertion. We’ll wait here.
Sure, go ahead, make sure you include the line where t = 1. Also, how wide is your pen?
This is a very cute story you tell … but now you have to convert your words into mathematical expressions …
So once we have done all of them, where is our pen? And don’t tell me “not defined”, because we are clearly still on the number line - after all, no step took us off of it. :rolleyes:
Am I understanding you correctly? You have some kind of rubric that separates real numbers from non-numbers and it involves moving a pen in discrete steps along a number line.
Can you define this procedure rigorously enough that others can use it and come to the same conclusions you do? Can I do this?
n = 1, t = 0: I move my pen from point 0 to point 0.999… of the number line
Why not?
You’ve hit the nail on the head. We’re all trying to demonstrate the absurdity. but we can’t tell which of us is getting through: Mr. Netz just keeps copy-pasting his rubric.
Let me try. I’ll demonstrate that zero is not a number.
At t = 0, my pencil is at 1.0
At t = 1 I move my pencil to 0.9
At t = 2 I move my pencil to 0.8
At t = 3 I move my pencil to 0.7
At t = 4 I move my pencil to 0.6 // think I’m going to make it? 
At t = 5 I move my pencil to Mars
At t = 6 I move my pencil to Jupiter
At t = 7 I move my pencil to Saturn
At t = 8 I move my pencil to …
Yeah but, the astute reader will notice that as ∞ ∉ ℕ then 0.999… doesn’t appear on the right hand side. That’s your problem, right there. That’s your example that is insufficient to model the case of an infinitely recurring decimal, that’s not a problem with maths, that’s a problem with your argument.
Just want to keep this one up there. I expect the answer is “An undefinable point undefinably close to 1”, but hope springs eternal.
You need him to define a number. He is apparently considering any symbol not defined/denoted by a fraction of integers to be a non number.
I’m pretty sure that you have accepted that 0.999… is greater than every element of the set {0.9,0.99,0.999, …} and whilst I cannot be arsed searching, I’m pretty sure you’ve held that 0.999… is indeed less than 1.
Still, if the whole of your argument is that your super-fine pen never reaches 0.999… therefore 0.999… is not well-defined, then you really have to explain why we should want to accept your bizarro definition of “well definedness” particularly because it excludes some very regular numbers, for example 0.333…, but yet somehow(?) allows for √2 and π.
His strawman keeps moving the goal posts … and Pluto is non-defined as an undefinable definition … wait for it … now we have Pluto is under-defined over the not-quite-defined non-definitions …
nm
True dat.
Mine, before I nm’ed it, was going to be
Why did nobody give me a “I see what you did there!”