I saw it and my inner 10-year old chuckled.
Where is the pen pointing to if it cannot have stopped at a specific point by definition? Infinitely many steps means NO STOPPING. You - and most others here - think it has stopped at 1. No one has shows how that happens - so far.
n = 1, t = 0: I move my pen from point 0 to point √-1 of the number line
Why not?
Or as septimus suggested: n = 1, t = 0: I move my pen from point 0 to point Saturn of the number line
Why not?
√-1 and Saturn do not correlate to the unit of the number line. π does (the ratio of a circle’s circumference to its diameter).
It is ok to reach the point 0.875 in a single step. Try this:
n = 1, t = 0: I move my pen from point 0 to point 0.875 of the number line
It should still work if you try to reach it in three steps - one step per digit:
n = 1, t = 0: I move my pen from point 0 to point 0.8 of the number line
n = 2, t = 0.5: I move my pen from point 0.8 to point 0.87 of the number line
n = 3, t = 1.0: I move my pen from point 0.87 to point 0.875 of the number line
That’s a problem of 0.999…
My list simply contains one step per digit in 0.999…
No. Saying it is at a point less than 1 is as wrong as saying it is at 1. The pen didn’t stop at all.
And now we’ve reached the point where you’re claiming it’s impossible to examine what happens after infinite steps in any way.
Would anyone else like to continue teaching this guy middle school mathematics? I think I’m done. The news this morning has officially sapped my willingness to take this kind of bullshit.
If it hasn’t stopped, we haven’t reached infinity. 0.999… isn’t a process, it isn’t a statement of movement, it definitely isn’t your specific description of movement.
You’ve here conclusively shown that the issue here is that you can’t understand the mathematical use of infinities. Which of course has been obvious for a long time …
Just for fun I’ll ask this question again since you’ve failed to resolve the issues it raises previously.
If I move the pen with constant speed its position at any time can be described with a number. These numbers have to include all numbers between 0 and 1 inclusive, including 0.99, 0.999, 0.999 and so on. There cannot be a point with a largest number of nines, which logically leads us to there being a point with infinite nines.
Forget about your step by step process for a second and look at the logic of this statement in isolation. Is there anything wrong with the premisses and deduction?
If there isn’t you simply cannot, by the concrete and irrefutable laws of logic, go “well I like my description better”. You must either declare you don’t believe in infinities or see that 0.999… can only be defined as the limit of the series.
Nothing happens after an infinite number of steps. The infinitely many steps alone define the state after an infinite number of steps.
Forget about your constant speed process. We are talking about 0.999… - one step for each digit in 0.999…
The constant speed situation is a valid mathematical construct that involves 0.999… Either show how it doesn’t, by proving either premisses or deduction wrong, or stop pretending you’re talking about mathematics.
Calculus proves (0.999…) = 1 … will you please show where this is in error.
So the list of pen moves is irrelevant? First you decide if it’s a number and if it’s not, you copy and paste a list of pen moves that don’t get there. Is that accurate?
Trouble is, you defined each of those steps to take half the time of the previous step. That means that time runs out after one second. The pen must stop then, because there isn’t any place it can move. There isn’t any time left.
You messed up: you defined a “supertask” that actually has to end.
I did copy and paste it several times. We need to identify a point 0.999… on the number line to be able to define the distance |1 - 0.999…|.
Can you rephrase your question, please? I will try to answer the first part of your question:
No matter which number of steps you plan to reach the target point you must succeed to reach it. If the claim is that we can reach a point 0.999… on the number line if we move 0.9 + 0.09 + 0.009 + … digit by digit, then the task is to define a procedure (a list) that shows how this point can be reached.
0.999… does define the supertask. 0.999… doesn’t define a stop at any point. It leaves the pen in an undefined state.
An “undefined state” that just happens to be infinitely close to 1 and not infinitely close to any other “defined” number.
The problem here is you want to introduce your own personal definition of what it means for a number to be defined. We can keep showing you how that doesn’t make sense, but as long as you operate with this nonsensical, if intuitively attractive, “axiom”, we’ll never agree.
It happens to be neither at 1 nor close to 1. It didn’t stop at any position < 1 either.
Then what you’re discussing isn’t math.
I haven’t read the thread. 
Did Mr. Netz ever indicate whether his argument against 0.99999 applies against 0.33333 ?
If yes, does 1/3 have a decimal representation, or is it just another Not_a_number?
If no, why is the pen moving from 0.9999 to 0.99999 not mimicked by the pen moving from 0.3333 to 0.33333?
Did anyone ever explain to Mr. Netz that 1 = 0.99999 ?