I see we’ve beaten this dead horse’s bones into flour now … shall we bake some bread?
No. There is an n-th digit for every n ∈ ℕ. This means infinitely many.
The error is when you go “none of the steps are 1, so it’s not 1 at infinity”. That’s treating infinity as a very big finite.
There is no infinitieth step on my list - just like there is no infinitieth digit in 0.999…
We know …
So, what do mean by “That’s treating infinity as a very big finite” then? There is no “infinitieth” digit, there is no “very big finite” digit. All there is is infinitely many finite digits, an n-th digit for every n ∈ ℕ. That’s well-defined.
If ∞ ∈ ℕ, then ℕ isn’t well defined at all. Infinity is an abstract concept describing something without any bound or larger than any number.
Ok. Thank you. That’s why I said [SIZE=“3”]n[/SIZE] ∈ ℕ, and not ∞ ∈ ℕ.
I agree, but this just means you’re using a very very large finite number of 9’s. So, you’d be correct to say that a decimal point followed by n (∈ ℕ) 9’s wouldn’t equal 1. The question is if we have 9’s forever then it does equal 1; such that:
1 - (0.999…) = (0.000…) = 0
Well, since there is no positive number that is less than 1 - (0.999…), it must equal zero.
No. I am not just using a “very very large finite number of 9’s”. See:
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
…
There is an n-th 9 added for each n ∈ ℕ. Tell me which finite number are you missing?
Then I can tell you where it is on the list. The list is complete - and still - there is no line of the list where I am moving the pen to point 1!
The question is, if the pen hasn’t been moved to point 1 within t = 1 second:
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
…
What is the position of the pen at t >= 1?
I don’t know if anyone’s said this yet, but the answer to your question is ONE.
Meanwhile back in reality :
you’ve asked and it’s been answered several times, so I’m wondering what you hope to gain by re-asking, and re-asking.
PS: when does the pen move from 0.9 to 0.99 ?
You can repeat and repeat it. As long as you don’t define which action on the infinite list does perform the move to point ONE it’s useless. It’s all about definitions.
Ok. Thanks. It should read:
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
…
t = 0: I move my pen from point 0 to point 0.0 of the number line
t = 0.0: I move my pen from point 0.00 to point 0.000 of the number line
t = 0.00: I move my pen from point 0.000 to point 0.0000 of the number line
…
I don’t know about you, but this looks like zero to me. At which n ∈ ℕ does this become non-zero? Then that would be your n that is just a very very large finite number.
Your whole argument is ridiculous, redundant, and already dealt with.
Usain Bolt runs at about 10 metres (1 decametre) per second (slightly faster, but suppose this is a bad day for him).
So :
at t = 0 he is at 0 decametres
at t = 0.9 he is at 0.9 decametres
at t = 0.99 he is at 0.99 decametres
at t = 0.999 he is at 0.999 decametres
at t = 0.9999 he is at 0.9999 decametres
Somehow you want to conclude that his position at time t = 1 isn’t defined, that he can never cover those ten metres. This is Zeno’s paradox, which has long since been resolved, and which doesn’t surprise anyone but flat-earthers, n00bs and trolls.
No, Zeno dealt with Achilles, not Usain Bolt. That’s clearly a completely different problem!
Infinities were Zeno’s Achilles’ heel.
This process doesn’t make sense to me. You should at least change it to:
t = 0: I move my pen from point 0 to point 0.0 of the number line
t = 0.5: I move my pen from point 0.00 to point 0.000 of the number line
t = 0.75: I move my pen from point 0.000 to point 0.0000 of the number line
…
Now we can see, that the position of the pen is 0 for every finite number, for every very very large finite number, and after completing all those infinitely many steps.
In the real world Usain Bolt does reach 1 at t = 1. But your list doesn’t define how he gets there. That’s it.