Heck, if we’re going to play that game, you can play it with just me.
#165 Zeno
#170 Zeno
#195 Zeno
#258 Zeno
#312 Zeno
#544 Zeno
Ta da.
Who cares? Important to know is where the center of the tip of the pen is pointing to.
What did he win?
He could also have asked what the limit of { 0.9, 0.99, 0.999, … } is. It’s simply 1.
No. There is no line “I move my pen from point 0.9 to point 0.9 of the number line”. It should appear at least once to guarantee that the pen is pointing to point 0.9.
No. My point is that there is no point 0.333… that can be reached - not even in infinitely many steps. Otherwise show how this happens.
t = 0: I move my pen from point 0 to point 0.3 of the number line
t = 0.9: I move my pen from point 0.3 to point 0.33 of the number line
t = 0.99: I move my pen from point 0.33 to point 0.333 of the number line
…
No action on this infinite list that shows that we stop at a specific point. Neither at 0.333… nor at 1/3. There are no actions other than those on this list.
Well, what that limit means is: start with 0 add a point and then nines forever after.
i.e. 0.999… = lim { 0.9, 0.99, 0.999, … } = 1.
So we’ve done here.
The other conclusion, that we would never have made if not for your graceful presence, is that one cannot physically do a thing forever.
And once again, SO WHAT?
t = 0: I move my pen from point 0 to point 0.1 of the number line
t = 0.9: I move my pen from point 0.1 to point 0.2 of the number line
t = 0.99: I move my pen from point 0.2 to point 0.3 of the number line
t = 0.999: I move my pen from point 0.3 back to point 0.2 of the number line
t = 0.9999: I move my pen from point 0.2 back to point 0.3 of the number line
t = 0.99999: I move my pen from point 0.3 back to point 0.2 of the number line
t = 0.999999: I move my pen from point 0.2 back to point 0.3 of the number line
…
No action on this infinite list that shows that we stop at a specific point. Neither at 0.333… nor at 7. There are no actions other than those on this list.
ETA: I once asked you to explain what your pen metaphor is about, and why we should care. But all you ever do is repeat the same nursery rhyme over and over and over.
…But 0.99… is defined as the limit of {0.9, 0.99, 0.999, … }.
You’re right, at least in part. If you start going:
0.3
0.33
0.333
…
then you’re correct that there’s an infinite number of steps that don’t get to 1/3.
But, since ALL of those steps have only a FINITE number of 3’s, this doesn’t say anything about the number 0.333~, which, loosely speaking, has an INFINITE number of 3’s. Get it?
I do … how wide is your pen … how wide is it’s center … how much of the number line is your pen pointing to ?
Absolutely correct. Here you agree that the distance between the left and the right point is not 0 at t = 1 because there is no line on the infinite list that shows that.
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 0.81 of the number line
t = 0.99: I move my pen from point 0.81 to point 0.9 of the number line
t = 0.999: I move my pen from point 0.9 to point 0.891 of the number line
t = 0.9999: I move my pen from point 0.891 to point 0.9 of the number line
t = 0.99999: I move my pen from point 0.9 to point 0.8991 of the number line
t = 0.999999: I move my pen from point 0.8991 to point 0.9 of the number line
t = 0.9999999: I move my pen from point 0.9 to point 0.89991 of the number line
…
Here you disagree that the distance between the left and the right point is not 0 at t = 1 although there is no line on the infinite list that shows that.
Why?
t = 0: I move my pen from point 0 to point 0.3 of the number line
t = 0.9: I move my pen from point 0.3 to point 0.33 of the number line
t = 0.99: I move my pen from point 0.33 to point 0.333 of the number line
…
By t = 1 the pen has moved all the distances { 0.3, 0.03, 0.003, … }. If there is a point 0.333… the pen should have reached it - simply by executing the actions on the list.
The center? It has no width.
You must be new around here.
LOL, point taken. Hope springs eternal. (But not for much longer.)
Well, there’s two different things going on here with what you’re saying. If you want to bring time into this and simply note where the pen is at any time, then yes the pen points to 0.333~ at t=1.
The other thing is that, no, you can’t get to 0.333~ by executing the actions on the list that we’ve already determined is an infinite list of points that fall short of the target, being partial sums. That assertion is simply false.
There, that should clear things up, for sure this time.
t = 0: I move my pen from 1 to 2
t = 0.5: I move my pen from 2 to 3
t = 0.75: I move my pen from 3 to 4
t = 0.875: I move my pen from 4 to 5
By t = 1, the pen has move all the distances in ℕ. If there is a point “infinity”… the pen should have reached it - simply by executing the actions on the list. Clearly, that means that when we finish the supertask of counting all the natural numbers, we won’t be at infinity, but rather at some natural number on the list.
Except that’s not how infinity works, and that’s not how limits work. The series {1, 2, 3…} clearly has a limit of infinity. If we “finish” the infinite series, we will, for lack of better terminology*, be at infinity… Even though infinity is not present at any step on that list. Because if you look only to the steps taken for an infinite process, you will miss the big picture.
*Okay technically it’s a divergent limit and there is better terminology here but I’m trying to explain the absolute basics to you here so we’ll be a little bit wrong to make a point.
Then it does not exist? If width = 0, then how does it point to anything?
You’re off to a GREAT start … I think that it will be you, alone, who can decipher this dilemma …
You’re missing the point. Literally.
0 dimensions means no length, no breadth, no height. No size whatsoever.
A non-zero-sized point would be like saying “kinda approximately 1.” If you mean 1 precisely, than the point defining 1 is a true zero-dimensional point.
Likewise, if you mean 3 * 1/3, you mean the precise zero-dimensional point which precisely defines 1. Or .9 (repeating infinitely). It’s the same exact point, with exactly the same dimensionality (zero).
This is not subject to debate. It’s fundamental mathematics. It’s not required to fit into your intuition in order to be correct.