An infinite question: Why doesn't .999~ = 1?

[watchwolf49]

@gnoitall

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.

Wrong context …

[gnoitall]

@watchwolf49

Wrong context …

Unless involving a pen is a wild hijack of the thread, a pen is a terrible analogue of a point, because it can’t be zero-dimensional.

So maybe I was yelling at the wrong person? Wouldn’t be the first time, or the last. :dubious:

The point, misdirected or not, stands. A precise number is a zero-dimensional point.

[watchwolf49]

@netzweltler

[snip] …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 … [snip]

This is just a snippet of something that has been posted in this thread well over a hundred times … we’re taking issue with this as a process … since as you correctly “point” out the number (0.999…) is in fact a point, and not a process.

Now, if you’re supporting Netzweltler’s position that (0.999…) ≠ 1 … then that’s a different conversation …

[John_W.Kennedy]

Can we bring in the infinite hotel at this point?

[netzweltler]

@jepflast

@netzweltler

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.

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.

Isn’t saying that I can’t get to 0.333… by moving the distances { 0.3, 0.03, 0.003, … } the same as saying that I can’t get to 0.333… by adding 0.3 + 0.03 + 0.003 + …?

@jepflast

There, that should clear things up, for sure this time.

[netzweltler]

@Budget_Player_Cadet

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.

If we finished the task by t = 1 we are neither at a point infinity nor at some natural number on the list. If you are at some natural number on the list you haven’t finished the task. So, you are at t < 1. The state of the pen is not defined for t ≥ 1.

@Budget_Player_Cadet

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.

[netzweltler]

@watchwolf49

Then it does not exist? If width = 0, then how does it point to anything?

If the center of the pen has no width the pen doesn’t exist?

[naita]

@netzweltler

Isn’t saying that I can’t get to 0.333… by moving the distances { 0.3, 0.03, 0.003, … } the same as saying that I can’t get to 0.333… by adding 0.3 + 0.03 + 0.003 + …?

Yes. So by that set of rules we can’t define 0.333… that way, but by way of logic (there is no last member of the set {0.3, 0.33, 0.333, 0.333…}) the number 0.333… must exist, and it’s quite obvious that it is equal to 1/3. To some people this shows that infinities must be treated as actual infinities, and that one has to find sensible axioms that give answers that fit with reality, others choose to harp on and on about pens moving fractions of a distance while refusing to give credit to Zeno.

[Telemark]

@netzweltler

If you are at some natural number on the list you haven’t finished the task.

Perhaps you’ve failed to notice that you’re talking about infinities and you concept of finishing a task doesn’t apply? As was said many pages ago, you’re still treating an infinite as a very large finite and as a result everything you propose is meaningless.

[Riemann]

We established 300 posts ago that netzweltler is simply reprising Zeno’s paradox, he is unwilling to accept that there have been any valid developments in mathematics since Ancient Greece, and for 300 posts he has obstinately ignored any input, just reiterating variations on Zeno’s paradox over and over again.

Einstein’s (apocryphal) definition of insanity: doing the same thing over and over again and expecting a different result.

Does anyone seriously doubt that if this thread reaches 3,000 posts, netzweltler will still just be repeating Zeno’s paradox over and over again?

Is this performance art, or what are you people trying to achieve here? You are starting to worry me.

[watchwolf49]

You’re starting to worry just now ??? …

We’re a group of peoples doing the same thing over and over again and expecting a different result. That makes all the difference in the world.

If the center of the pen has no width the pen doesn’t exist?

No pen made of molecules … we’d expect the tip to be at least as wide as an electron … very very small but still non-zero …

[Trinopus]

@John_W.Kennedy

Can we bring in the infinite hotel at this point?

It’s been mentioned already a couple of times.

“We’ll leave infinity on for ya.”

[netzweltler]

@naita

@netzweltler

Isn’t saying that I can’t get to 0.333… by moving the distances { 0.3, 0.03, 0.003, … } the same as saying that I can’t get to 0.333… by adding 0.3 + 0.03 + 0.003 + …?

Yes.

So, 0.3 + 0.03 + 0.003 + … ≠ 0.333…
Do you mean both is true or both is wrong?

@naita

So by that set of rules we can’t define 0.333… that way, but by way of logic (there is no last member of the set {0.3, 0.33, 0.333, 0.333…}) the number 0.333… must exist, and it’s quite obvious that it is equal to 1/3. To some people this shows that infinities must be treated as actual infinities, and that one has to find sensible axioms that give answers that fit with reality, others choose to harp on and on about pens moving fractions of a distance while refusing to give credit to Zeno.

[netzweltler]

@Telemark

Perhaps you’ve failed to notice that you’re talking about infinities and you concept of finishing a task doesn’t apply? As was said many pages ago, you’re still treating an infinite as a very large finite and as a result everything you propose is meaningless.

The task is finished by t = 1. And of course it applies because we are talking about infinity.

[TriPolar]

I keep trying to read all the way through this thread but it keeps getting longer.

[netzweltler]

@watchwolf49

No pen made of molecules … we’d expect the tip to be at least as wide as an electron … very very small but still non-zero …

Saying that the center of the pen has some size is like saying that the center of gravity of the pen has a minimum size.

[Riemann]

@TriPolar

I keep trying to read all the way through this thread but it keeps getting longer.

Yes, but posts #101-#596 only added 3 more bytes of complexity.

[Telemark]

@netzweltler

The task is finished by t = 1. And of course it applies because we are talking about infinity.

Talking about infinity doesn’t mean you understand infinity. Which you clearly don’t.

[Bullitt]

@TriPolar

I keep trying to read all the way through this thread but it keeps getting longer.

@Riemann

Yes, but posts #101-#596 only added 3 more bytes of complexity.

It will be a long thread, but finitely long.
(Shhh, don’t tell that to netzweiler. And, we haven’t covered the concepts of countably infinite, and uncountably infinite, either.)

[naita]

@netzweltler

So, 0.3 + 0.03 + 0.003 + … ≠ 0.333…
Do you mean both is true or both is wrong?

Neither is mathematically rigorous. “Get to” requires a different approach for infinities than finites. In your process you claim we can’t get to 0.999…, and occasionally you realise that then we can’t get to t=1 either. But then you turn around and talk about the state at t=1 like now.

It is a restating of zeno’s paradoxes. Your “process” can’t reach t=1, except by accepting one of the solutions to Zeno’s paradoxes and acknowledging that 0.999… = 1. A different approach is required. If any of us had an sense we’d just repeat that last sentence from now on.