Simulation of smoothness using discrete quantities

[ryan]

Where I left off in my personal dealings with infinity is that there isn't really an infinite number of "halves" that the arrow has to travel to reach its end point. It goes through an arbitrarily large natural number of halves (whatever that means, and I believe this is the formal way of putting it).

So it seems as if it's something that isn't quite infinity but is larger than any fixed number.

Wut? It's pretty clear, in this case, that 1/2-1/2 + 1/3-1/3 + 1/4-1/4 ... = 0+0+0.... = 0

What are you talkin bout?

I didn't say it wasn't 0 (isn't it anything?).

I was just responding to steve, no derail intended.

 

[steve_bank]

1+1/2+1/3+1/4+1/5... - [1/2 +1/3 +1/4 + 1/5...] = 1

Then I would have the same issue as om the infinity threads. The idea that an infinite summation of finite numbers can lead to a finite result. A staement, not opening debate, but I would ssay the series approches 1 as a limit.

The Riemann series theorem doesn't really apply here, as I'm assuming a 1 to 1 mapping of 1/2 -1/2 + 1/3-1/3 + 1/4-1/4+... which is essentially an infinite amount of 0s added up ~ 1 + 0 + 0+ 0+ 0+ 0+0...=1

If you want to rearrange the terms, you're playing fast and loose with the rules, to quote someone with a bit more education than me.

Enough said, thanks.

 

[Keith&Co.]

in my personal dealings with infinity is that there isn't really an infinite number of "halves" that the arrow has to travel to reach its end point.

Then what stops it?

If you keep dividing the remaining distance to the target by halves, what forces you to stop doing that?

It stops somewhere between an arbitrarily large number and infinity, where ever that is.

I didn't ask where, i asked whst stops it?

 

[ryan]

It stops somewhere between an arbitrarily large number and infinity, where ever that is.

I didn't ask where, i asked whst stops it?

If I knew what, I would be famous. But the where is good enough for thos century.

 

[bilby]

"something that isn't quite infinity but is larger than any fixed number" is handwaving nonsense. It's self contradicting.

A number that "is larger than any fixed number" is infinite, by definition. So you are resolving your inability to comprehend infinity, by invoking an infinity that isn't infinite; And you find this somehow an IMPROVEMENT?? That's seriously fucked up right there.

 

[ryan]

"something that isn't quite infinity but is larger than any fixed number" is handwaving nonsense. It's self contradicting.

A number that "is larger than any fixed number" is infinite, by definition. So you are resolving your inability to comprehend infinity, by invoking an infinity that isn't infinite; And you find this somehow an IMPROVEMENT?? That's seriously fucked up right there.

So then explain why we say "goes to infinity" but is never infinity. For example sum of 1/2 + 1/4 ... = 1 "as n goes to infinity". I have been told by people with higher math levels than me that n does not equal infinity.

So we are left somewhere in between a fixed natural number and infinity.

 

[bilby]

"something that isn't quite infinity but is larger than any fixed number" is handwaving nonsense. It's self contradicting.

A number that "is larger than any fixed number" is infinite, by definition. So you are resolving your inability to comprehend infinity, by invoking an infinity that isn't infinite; And you find this somehow an IMPROVEMENT?? That's seriously fucked up right there.

So then explain why we say "goes to infinity" but is never infinity. For example sum of 1/2 + 1/4 ... = 1 "as n goes to infinity". I have been told by people with higher math levels than me that n does not equal infinity.

So we are left somewhere in between a fixed natural number and infinity.

Because n goes from your starting point (in this case, 1) to infinity. At n=infinity the value of the item in the series 1/2 + 1/4 + 1/8 ... 1/n² would be zero - if you get to n=infinity, you have evaluated every non-zero term in the infinite series. But the sum for ALL n, as n goes to infinity is 1.

n does not equal any natural number, nor does it equal infinity. Rather, it is a placeholder for every natural number.

 

[Keith&Co.]

It stops somewhere between an arbitrarily large number and infinity, where ever that is.

I didn't ask where, i asked whst stops it?

If I knew what, I would be famous. But the where is good enough for thos century.

So, you're just bullshitting.
I mean, there should be a reason you assume there's alimit in there somewhere...

 

[Kharakov]

"something that isn't quite infinity but is larger than any fixed number" is handwaving nonsense. It's self contradicting.

A number that "is larger than any fixed number" is infinite, by definition. So you are resolving your inability to comprehend infinity, by invoking an infinity that isn't infinite; And you find this somehow an IMPROVEMENT?? That's seriously fucked up right there.

So then explain why we say "goes to infinity" but is never infinity. For example sum of 1/2 + 1/4 ... = 1 "as n goes to infinity". I have been told by people with higher math levels than me that n does not equal infinity.

So we are left somewhere in between a fixed natural number and infinity.

It's the limit- in other words, you can think of the number line between 0 and 1 as completely saturated with the summation of 2^-ns in the case you mention. There is no smallest 2^-n that is not present- all gaps are filled.

 

[ryan]

If I knew what, I would be famous. But the where is good enough for thos century.

So, you're just bullshitting.
I mean, there should be a reason you assume there's alimit in there somewhere...

Well sure, call it what ever you want. We do know that it stops at 1. Steve seems to have an issue with that which is why I brought all this up in the first place.

- - - Updated - - -

"something that isn't quite infinity but is larger than any fixed number" is handwaving nonsense. It's self contradicting.

A number that "is larger than any fixed number" is infinite, by definition. So you are resolving your inability to comprehend infinity, by invoking an infinity that isn't infinite; And you find this somehow an IMPROVEMENT?? That's seriously fucked up right there.

So then explain why we say "goes to infinity" but is never infinity. For example sum of 1/2 + 1/4 ... = 1 "as n goes to infinity". I have been told by people with higher math levels than me that n does not equal infinity.

So we are left somewhere in between a fixed natural number and infinity.

Because n goes from your starting point (in this case, 1) to infinity. At n=infinity

n must be a natural number in this case. Infinity is not a natural number.

 

[ryan]

"something that isn't quite infinity but is larger than any fixed number" is handwaving nonsense. It's self contradicting.

A number that "is larger than any fixed number" is infinite, by definition. So you are resolving your inability to comprehend infinity, by invoking an infinity that isn't infinite; And you find this somehow an IMPROVEMENT?? That's seriously fucked up right there.

So then explain why we say "goes to infinity" but is never infinity. For example sum of 1/2 + 1/4 ... = 1 "as n goes to infinity". I have been told by people with higher math levels than me that n does not equal infinity.

So we are left somewhere in between a fixed natural number and infinity.

It's the limit- in other words, you can think of the number line between 0 and 1 as completely saturated with the summation of 2^-ns in the case you mention. There is no smallest 2^-n that is not present- all gaps are filled.

They say that the number of elements in the set of natural numbers is the number aleph null (I have read in numerous sources that aleph null is not just the cardinality of a countable set, N in this case, but is also a number.

Now let me try a proof by contradiction that there can only be a natural number of elements in the set of natural numbers.

Let S be a set that is an increasing sequence of natural number starting from n = 1. Let m be the number of elements in S and let n = m.

n is an element of the naturals, but m is undefined.

For example, when m = 5, we know that there are 5 elements, and according to the rule, there must be an n = 5 in S.

But what happens when we make m = aleph null, which is supposedly the number required to fill S with all N. n cannot equal aleph null because only natural numbers can be in the set of all natural numbers N, or in this case S.

The overall point is that the number of elements in N does not seem to be enough to match each natural number. Aleph null seems to have more elements than N.

 

[Kharakov]

Looks gibberishy to me ryan. Infinity means without bound. It is not a specific number. WTF you talkin bout willis?

 

[Keith&Co.]

Well sure, call it what ever you want. We do know that it stops at 1. Steve seems to have an issue with that which is why I brought all this up in the first place.

It stops at one because each time the arrow travels another 'half' its speed-per-half doubles. As the number of halves rises to infinity, the speed it covers each half approaches infinity. So the sequence of halves never ends, they're just summed up fairly quickly.

 

[ryan]

Looks gibberishy to me ryan. Infinity means without bound. It is not a specific number. WTF you talkin bout willis?

They are called transfinite numbers. https://www.britannica.com/science/transfinite-number

We see that when m = 3, where S would have a cardinality of 3 by the given rule for S, n maxes out at n = 3.

But if we claim that there are aleph null elements in S, then it forces there to be an n = "a transfinite number", which is not a natural number.

 

[Kharakov]

Ok, so you're saying the rules of your set are that n is both transfinite and natural? Pick one ryan.

Something can't both be completely nonexistent and exist.

 

[ryan]

Well sure, call it what ever you want. We do know that it stops at 1. Steve seems to have an issue with that which is why I brought all this up in the first place.

It stops at one because each time the arrow travels another 'half' its speed-per-half doubles. As the number of halves rises to infinity, the speed it covers each half approaches infinity. So the sequence of halves never ends, they're just summed up fairly quickly.

But like you asked, at what point does it go at a finite speed to an infinite speed?

 

[ryan]

Ok, so you're saying the rules of your set are that n is both transfinite and natural? Pick one ryan.

Something can't both be completely nonexistent and exist.

No, that is the outcome of my proof by contradiction. n* would have to be transfinite to count each n in N, thus the contradiction.

 

[bilby]

Well sure, call it what ever you want. We do know that it stops at 1. Steve seems to have an issue with that which is why I brought all this up in the first place.

- - - Updated - - -

"something that isn't quite infinity but is larger than any fixed number" is handwaving nonsense. It's self contradicting.

A number that "is larger than any fixed number" is infinite, by definition. So you are resolving your inability to comprehend infinity, by invoking an infinity that isn't infinite; And you find this somehow an IMPROVEMENT?? That's seriously fucked up right there.

So then explain why we say "goes to infinity" but is never infinity. For example sum of 1/2 + 1/4 ... = 1 "as n goes to infinity". I have been told by people with higher math levels than me that n does not equal infinity.

So we are left somewhere in between a fixed natural number and infinity.

Because n goes from your starting point (in this case, 1) to infinity. At n=infinity

n must be a natural number in this case. Infinity is not a natural number.

You are correct; But somehow I feel that you think this fact is of some importance. I am not sure why you would think that, other than that you stopped reading too soon, and cut off the rest of my comment. I very clearly continued with "would be zero", a hypothetical statement, about an imaginary circumstance. I wasn't proposing that n ever actually has a value of 'infinity'.

n goes to infinity - there is no largest value for n; There is no largest natural number. No matter what natural number you pick as a possible n_max, there are an infinity of even larger natural numbers. Hence 'goes to' and not 'is', which was your original question.

 

[bilby]

Well sure, call it what ever you want. We do know that it stops at 1. Steve seems to have an issue with that which is why I brought all this up in the first place.

It stops at one because each time the arrow travels another 'half' its speed-per-half doubles. As the number of halves rises to infinity, the speed it covers each half approaches infinity. So the sequence of halves never ends, they're just summed up fairly quickly.

But like you asked, at what point does it go at a finite speed to an infinite speed?

It never does; Speed is distance divided by time, and the time required to travel the remaining distance falls in exact proportion to the distance left to travel. The speed is constant - it never approaches infinity, and indeed it never increases above, nor falls below, its original value.

 

[ryan]

But like you asked, at what point does it go at a finite speed to an infinite speed?

It never does; Speed is distance divided by time, and the time required to travel the remaining distance falls in exact proportion to the distance left to travel. The speed is constant - it never approaches infinity, and indeed it never increases above, nor falls below, its original value.

I think he meant segments (or halves) per time. At least that was what I was answering to.