It’s not a process, it’s a number … there’s a difference …
You’ve previously insisted that the sequence on the left (0, 0.9, 0.99, … ) NEVER reaches 1.
I’m taking it as a victory that you have finally seen the folly of your ways.
Hurrah for persuasion! Maths is once more saved and the world can continue spinning safely on its axis.
Obviously you are talking about the sequence marked in red. t means the time when an action takes place. This doesn’t mean that time doesn’t reach 1, 2, and after. There are just no actions specified for these times.
So, you’re just picking and choosing which numbers your process works, and ignoring all the rest … does that sound mathematical to you? I’m pretty sure you’re missing an important step, you still have to integrate the whole shmear (from 0 to ∞). Perhaps Riemann would be kind enough to explain …
I am picking any numbers which make my process work. Any sequence which guarantees that I have completed infinitely many steps by t = 1. It can be t = { 0, 0.9, 0.99, 0.999, … }. It can also be t = { 0, 0.5, 0,75, 0.875, … }.
No step missing.
Except it obviously doesn’t work, or at least doesn’t work they way you claim it does. If you’ve completed infinitely many steps by the time, and only by the time, you reach t=1, then 0.999… = 1
Note the bit in bold. If you have any difference between the “almost infinite” number of nines after the decimal point and 1, then you haven’t reached t=1.
As I said once before, you’re free to decide that infinites are impossible, or that there’s an infinitesimal “last step”, but that is not some deep mathematical truth you have discovered. It’s a choice and a poor one. For instance it means that for your process where the value is 0.9 when t=0.9 and 0.99 when t=0.99 you cannot reach 1.
At times it seems like that is your point, you cannot reach t=1, but then you post something like the above, where you say you’ve completed infinitely many steps by t=1. So which is it? And try to give it some thought and examine where we might be misunderstanding you, if anywhere.
We are reaching t = 1. Time continues. I was stating, we don’t reach point 1 as a part of this process:
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 0.9 to point 0.99 of the number line
t = 0.99: I move my pen from point 0.99 to point 0.999 of the number line
…
Even when we have reached t = 1 we haven’t reached point 1. Because all the action takes place at t < 1. There is no action specified for t = 1 and after.
Yes. It’s where 1 is.
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Is this argument still going on? The correct and incorrect answers have each been given multiple times. Doesn’t it (like 0.999999… :rolleyes: ) get repetitious?
Let me help! 
Let’s start with this, Zeno’s paradox of Achilles and the tortoise.
@ netzweltler — Do you agree that “In a race, the quickest runner can never overtake the slowest”? If not, tell us what is wrong (or right) with Zeno’s reasoning. When you complete this assignment, connect the dots and relate it to the topic of this thread.
That’s what I thought. So let’s describe a different process, that is similar to yours. I move my pen at a constant speed of one unit on the number line per one unit of time.
Now the “steps” below aren’t steps, they’re just statements of facts about some of the states as we complete the process above.
At t=0 I’m at 0
At t=0.9 I’m at 0.9
At t=0.99 I’m at 0.99
…
At t=1 I’m at 1.
The starting points of every step in your process has an equivalent state in this process, and in this process we do reach 1, so at some point we passed a point that could be described by infinite nines after the decimal point.
What is your opinion about this process? That we didn’t pass a point that could be described by infinite nines? (What’s the maximum number of nines then?) That that point can’t be defined as anything? (But we did pass it, and we can give a really good description of when.) That that point isn’t 1 because of your step by step process? (But according to you your process never reaches 1, how can it say anything about 1 when it doesn’t reach it?)
Now please don’t answer this following statement unless it’s to accept it’s correct. We already know you disagree with it, so you need to focus on the paragraph above if we’re to stop going in circles. But my, and all of mathematics, claim, is that we only pass that point of infinite nines when t=1. At t=0.9 there is 0.1 time units left until we pass 1, at t=0.99 there is 0.01 time units left, at t=0.999 there is 0.001 time units left. As we approach 0.999…, the one with infinite nines, which must exist by this description of things, we approach 0.000… infinite zeros. With infinite zeros there is no terminating 1, so 0.000… is properly and exactly 0, which means that 0.999… is reached at t=1, which is also when we’re at 1, so 0.999… is equal to 1.
Good luck there, I already tried, he won’t be convinced.
For simplification we should deal with the dichotomy paradox and not the paradox of Achilles and the tortoise in this context.
As you say, your process is not dealing with steps. So yes, when continuously moving you reach point 1 after passing the infinitely many 9s in 0.999…
But that’s not what’s going on when you add 0.9 + 0.09 + 0.009 + … We are dealing with steps!
It can, because my process doesn’t specify an action for t = 1. So, nothing happens at t = 1. Therefore, the state of the pen is solely defined by the actions at t < 1. At t = { 0, 0.9, 0.99, 0.999, … }.
No. You’re dealing with steps. The rest of the world recognizes that you can’t do that when discussing infinities. It’s pointless and non-productive to do so. You can continue to play this game and get no useful answers or explain why you think your thought experiment is meaningful.
Try reading my description again. All your steps exist as states in my process. Do you agree with that?
And I’m not saying we reach point 1 after passing infinitely many 9s, we reach point 1 when we reach infinitely many 9s.
Do you agree that in the constant speed process the difference between 0.999… and 1 is 0? If not, what is the difference?
Wrong. Achilles and the tortoise is the perfect paradox to mimic your thoughts. Answer the questions please.
0.999… is dealing with steps. Infinitely many of them. And this means discussing infinity, of course.
It was meaningful for centuries. Since Zeno came up with that for the first time.
Yes. So what?
There is no point when we reach infinitely many 9s. What should that be? The last point?
0.999… is not a result of a constant speed process.
Sorry. Still cannot see how this paradox is related to the problem under discussion. Please explain.
So the constant speed process encompasses your step by step process.
If you’d paid any intention you’d know that I already consider that point 1. But think about what you’re saying here. Moving from 0 to 1 at a constant speed we pass every possible point on the number line in the interval [0, 1] in time t=[0, 1], this includes 0.9, 0.99, 0.999 and so on. If it doesn’t include infinite 9’s, what’s the largest number of nines it includes?
0.999… doesn’t have to be the result of any process, it’s a number in and of itself. And as I just described logic entails it occurs as a state in the constant speed process. Try to look outside the box you’ve got yourself stuck in here. Think about the process I just described. We’re moving from 0 to 1 on a number line over the span of 1 unit of time. At 0.9 we’re at 0.9, at 0.99 we’re at 0.99 and so on. Unless we can point to a maximum number of nines, during this movement there has to be a point we can describe by an infinite number of nines. Where is that point, and if you think it doesn’t exist, what is the largest number of nines describing a point we do pass by.
The two formulations seem extremely similar:
Achilles never reaches the tortoise because his residual distance (1/10, 1/100, 1/1000, …) remains positive after any finite number of steps.
Netzwelter never reaches 1/3 because his residual distance (0.333…, 0.0333, 0.00333…, 0.0003333 …) remains positive after any finite number of states.