(March 2, 2018 at 2:53 pm)RoadRunner79 Wrote: Well, I explained my reasoning for a last point (it's not an assumption).  Even with only two points on a segment if we start at zero and end at A, then zero would be the last point before A.  With more than one point and progression to A through these points, you will have a last point.  It is not dependent on time at or between points, as long as there is some sort of movement towards the destination.

However If you are not going to engage, and just keep repeating the claim, then I'm not going to repeat myself..... Now after numerous posts you admit that this process which shows your infinite points also prevents you from reaching the destination at all.  There is some mystery process which is added, and allows us to end the other process, which is being defined as infinite (without end or limit).    If you have something new to talk about, or wish to engage with something that I have said... I'm happy to discuss.   Otherwise I'll leave the last word to you if you desire.

And if there are only finitely many points between, then there will be a last point. But if there are infinitely many, there does not have to be. That is simply one of the ways infinite sets differ from finite ones: infinite ones need not have a first or last.

The underlying assumption you seem to make is that an infinite number of steps cannot be finished. But they can, *if* they are done faster and faster, so that the total time is still finite.

So, the difficulty seems to be that an infinite sum of positive quantities can be finite. it is easy to see that they can be bounded. For example:

.9
.99
.999
.9999
.99999
...

I think you will agree that there are infinitely many numbers in this list (whether potential or actual). But, clearly, every number on this list is smaller than 2.

So this infinite set (potential, if you like) is bounded above by 2.

This is an example of an infinite, but bounded collection. Even if you only want to do 'potential' infinities, the numbers that appear are ALL smaller than 2 (and, in fact, each is smaller than 1).

So, even with a potential infinity, it is possible to be infinite and bounded. It is infinite in quantity, but finite in terms of size of the numbers involved.

And, if these represent time passage, we DO go through all of them and reach 1. Not by staying in this sequence, but by going through the sequence in a finite time.

(March 2, 2018 at 4:34 pm)polymath257 Wrote:

(March 2, 2018 at 2:53 pm)RoadRunner79 Wrote: Well, I explained my reasoning for a last point (it's not an assumption).  Even with only two points on a segment if we start at zero and end at A, then zero would be the last point before A.  With more than one point and progression to A through these points, you will have a last point.  It is not dependent on time at or between points, as long as there is some sort of movement towards the destination.

However If you are not going to engage, and just keep repeating the claim, then I'm not going to repeat myself..... Now after numerous posts you admit that this process which shows your infinite points also prevents you from reaching the destination at all.  There is some mystery process which is added, and allows us to end the other process, which is being defined as infinite (without end or limit).    If you have something new to talk about, or wish to engage with something that I have said... I'm happy to discuss.   Otherwise I'll leave the last word to you if you desire.

And if there are only finitely many points between, then there will be a last point. But if there are infinitely many, there does not have to be. That is simply one of the ways infinite sets differ from finite ones: infinite ones need not have a first or last.

The underlying assumption you seem to make is that an infinite number of steps cannot be finished. But they can, *if* they are done faster and faster, so that the total time is still finite.

So, the difficulty seems to be that an infinite sum of positive quantities can be finite. it is easy to see that they can be bounded. For example:

.9
.99
.999
.9999
.99999
...

I think you will agree that there are infinitely many numbers in this list (whether potential or actual). But, clearly, every number on this list is smaller than 2.

So this infinite set (potential, if you like) is bounded above by 2.

This is an example of an infinite, but bounded collection. Even if you only want to do 'potential' infinities, the numbers that appear are ALL smaller than 2 (and, in fact, each is smaller than 1).

So, even with a potential infinity, it is possible to be infinite and bounded. It is infinite in quantity, but finite in terms of size of the numbers involved.

And, if these represent time passage, we DO go through all of them and reach 1. Not by staying in this sequence, but by going through the sequence in a finite time.

You may do things as fast or slow as you like.   In Zeno's dichotomy paradox, and your X<Y<1    no matter how much time, you will never reach 1, you will always have another point to go through that is less than 1 (hence why the term infinity applies). This is not an assumption... this is the math function being described!  The argument is not that it will take an extraordinary amount of time to complete!

(March 2, 2018 at 5:09 pm)RoadRunner79 Wrote:

(March 2, 2018 at 4:34 pm)polymath257 Wrote: And if there are only finitely many points between, then there will be a last point. But if there are infinitely many, there does not have to be. That is simply one of the ways infinite sets differ from finite ones: infinite ones need not have a first or last.

The underlying assumption you seem to make is that an infinite number of steps cannot be finished. But they can, *if* they are done faster and faster, so that the total time is still finite.

So, the difficulty seems to be that an infinite sum of positive quantities can be finite. it is easy to see that they can be bounded. For example:

.9
.99
.999
.9999
.99999
...

I think you will agree that there are infinitely many numbers in this list (whether potential or actual). But, clearly, every number on this list is smaller than 2.

So this infinite set (potential, if you like) is bounded above by 2.

This is an example of an infinite, but bounded collection. Even if you only want to do 'potential' infinities, the numbers that appear are ALL smaller than 2 (and, in fact, each is smaller than 1).

So, even with a potential infinity, it is possible to be infinite and bounded. It is infinite in quantity, but finite in terms of size of the numbers involved.

And, if these represent time passage, we DO go through all of them and reach 1. Not by staying in this sequence, but by going through the sequence in a finite time.

You may do things as fast or slow as you like.   In Zeno's dichotomy paradox, and your X<Y<1    no matter how much time, you will never reach 1, you will always have another point to go through that is less than 1 (hence why the term infinity applies). This is not an assumption... this is the math function being described!  The argument is not that it will take an extraordinary amount of time to complete!

Let me do it like this. Suppose you are attempting to go through the sequence .9, .99, .999, .9999, .99999, .... There are two scenarios..

In the first, you are at .9 after 1 second, at .99 after 2 seconds, .999 after 3 seconds, etc. I agree that at this rate you will never get through the sequence: it will take an infinite amount of time to do so.

In the second, though, you are at .9 after 1 second, .99 after 1.5 seconds, .999 after 1.75 seconds, .9999 after 1.875 seconds, etc. Each step takes half the time of the previous one. In this scenario, you *will* get through the sequence of 9's after 2 seconds. At the 2 second mark you are past every single one of them.

And yes, whenever you are at any of those points, you have more to go through.
And yes, when you get to t=2, you are past all of them.
And no, there isn't a last one you went through.
And yes, you went through all of them.

The point: you do, in fact, go through an infinite number of times in every finite duration.

(March 2, 2018 at 6:48 pm)polymath257 Wrote:

(March 2, 2018 at 5:09 pm)RoadRunner79 Wrote: You may do things as fast or slow as you like.   In Zeno's dichotomy paradox, and your X<Y<1    no matter how much time, you will never reach 1, you will always have another point to go through that is less than 1 (hence why the term infinity applies). This is not an assumption... this is the math function being described!  The argument is not that it will take an extraordinary amount of time to complete!

Let me do it like this. Suppose you are attempting to go through the sequence .9, .99, .999, .9999, .99999, .... There are two scenarios..

In the first, you are at .9 after 1 second, at .99 after 2 seconds, .999 after 3 seconds, etc. I agree that at this rate you will never get through the sequence: it will take an infinite amount of time to do so.

In the second, though, you are at .9 after 1 second, .99 after 1.5 seconds, .999 after 1.75 seconds, .9999 after 1.875 seconds, etc. Each step takes half the time of the previous one. In this scenario, you *will* get through the sequence of 9's after 2 seconds. At the 2 second mark you are past every single one of them.

And yes, whenever you are at any of those points, you have more to go through.
And yes, when you get to t=2, you are past all of them.
And no, there isn't a last one you went through.
And yes, you went through all of them.

The point: you do, in fact, go through an infinite number of times in every finite duration.

Why does this not work when you do the same process that you used to show an infinity, and is also the one in Zeno’s paradox? You always go to something different and don’t address this in the dichotomy paradox.

Note for everyone else. You may notice, that the math in Zeno’s paradox will never reach it’s goal. If the math gives you trouble, you can see that this same process is being used to show that there is an infinite amount of numbers, that you cannot get to the destination, then you add another infinite, and can then reach the destination. In more ways than one... what is being claimed simply does not add up.

(March 2, 2018 at 6:59 pm)RoadRunner79 Wrote:

(March 2, 2018 at 6:48 pm)polymath257 Wrote: Let me do it like this. Suppose you are attempting to go through the sequence .9, .99, .999, .9999, .99999, .... There are two scenarios..

In the first, you are at .9 after 1 second, at .99 after 2 seconds, .999 after 3 seconds, etc. I agree that at this rate you will never get through the sequence: it will take an infinite amount of time to do so.

In the second, though, you are at .9 after 1 second, .99 after 1.5 seconds, .999 after 1.75 seconds, .9999 after 1.875 seconds, etc. Each step takes half the time of the previous one. In this scenario, you *will* get through the sequence of 9's after 2 seconds. At the 2 second mark you are past every single one of them.

And yes, whenever you are at any of those points, you have more to go through.
And yes, when you get to t=2, you are past all of them.
And no, there isn't a last one you went through.
And yes, you went through all of them.

The point: you do, in fact, go through an infinite number of times in every finite duration.

Why does this not work when you do the same process that you used to show an infinity, and is also the one in Zeno’s paradox?  You always go to something different and don’t address this in the dichotomy paradox.

Note for everyone else. You may notice, that the math in Zeno’s paradox will never reach it’s goal. If the math gives you trouble, you can see that this same process is being used to show that there is an infinite amount of numbers, that you cannot get to the destination, then you add another infinite, and can then reach the destination. In more ways than one... what is being claimed simply does not add up.

OK, again, part of the problem is the definition of the term 'infinity'. You claim an infinity cannot have an end and that it cannot be bounded. That is simply false. The sequence of 9's above is a wonderful example.

For each one in the sequence, there are an infinite number that are larger. This is true. So it is an infinite collection (whether potential or actual is irrelevant here).

It is also bounded since everything in that sequence is less than 1. Each and every term of that sequence is smaller than 1.

So, we have an infinity that is bounded.

The resolution is easy: we have two different notions of boundedness here. One has to do with where in the e a term is and the other has to do with where in the interval from 0 to 1 that term is. The sequence is unbounded in the first sense and bounded in the second. But they are two different notions of being bounded, so there is no contradiction there.

When you ask if we can 'get through' a sequence, you have to specify exactly what you mean. Strictly speaking there isn't even a process here. There are two collections: the terms of the sequence and the numbers between 0 and 1. Both are infinite. You don't get to 1 by staying in the sequence. But that isn't relevant if you want to get to one from within the numbers between 0 and 1.

But we *do* get to the destination, which means we *do* go through all those points. If you deny that, exactly which point do we not go through? At which point does the division break down?

Quote:Mathematicians are an interesting bunch. They are very, very rigorous when it comes to analyzing implications – what follows from what. They do not seem nearly as rigorous when it comes to analyzing presuppositions – what precedes from what. In fact, they do not even seem to be aware of their own presuppositions. I’ve been told countless times, “It’s absolutely certain that Cantor proved the existence of different sizes of infinite sets! Mathematicians have double-checked his work for a century!”

But they don’t seem to be aware of one problem: what if the presuppositions of Cantor’s proof are wrong? What if – specifically – the concepts that he presupposed were imprecise.

Great, so disprove these "presuppositions". Show us how they're wrong. Steve Patterson can say whatever he wants, but even he has the burden to show that the concepts are problematic.

---

(March 2, 2018 at 12:28 pm)SteveII Wrote: This is the type of article I have asked from Grandizer and Polymath for 40 pages. Where are the philosophy articles that defend an actual infinite in the real world?

Physicists not good enough for you anymore? No problem. Anytime you read about Cantor's set theory, you are implicitly reading about a philosophy that defends an actual infinity. Also, it looks like you've already forgotten about Wes Morriston. Didn't you quote him earlier somewhere?

(March 2, 2018 at 11:02 am)SteveII Wrote: Since you took the time to kind of reply to my syllogism...

You are incapable of separating the concept of a mathematical infinity with one in real life or a logically possible infinity. That's why you think this question has been answered. I have shown conclusively that the concept of infinity in mathematics is not the same. It was not derived from a logical process. It is assumed for the purpose of further math calculations. Period. End of story. If you use the mathematical concept of infinity as an argument against anything related to infinity with real objects (like Hilbert's Hotel), you are question begging. By one of your previous comments, you seem to think this is okay to do. It is NOT.

I love how when you're incapable of demonstrating that an actual infinity is logically impossible, you charge me with question begging as if I am presuming the logical possibility of an actual infinity to prove an actual infinity. What I'm doing is presuming the logical possibility of actual infinity and see if it leads to a logical contradiction. And since there is no contradiction that has been pointed out thus far, then it makes sense to say that it seems logically possible, and that there is no reason to argue that it's logically impossible. Are you noting the subtle differences here?

Quote:For the 40th time, all you have to do is produce an article that talks about the possibility of an actual infinity. This does not mean an article that mentions a multiverse. That is not a discussion on the concept of infinity (the subject of this thread). I am talking about an article that discusses the concept. This should be easy if I am "mindlessly repeating the same shit which has been refuted regardless".

Read any article on Cantor's set theory, and you have your philosophical article right there.

Quote:Why do I keep asking for an article? Because, no offense, the nuances of this subject are beyond you. You stumble along with complete confidence that is not warranted by your piss-poor critical thinking, argumentation and discussion skills. I am not usually as blunt, but your cockiness and condescension is getting to me.

Nice projection there, LOL. Weren't you the one, by the way, that mistakenly thought a question-begging argument was an invalid argument? Some critical thinking skills you have there?

Quote:I will clarify the obvious:

1' An actual infinite in the physical world consists of real (not abstract) objects.

That depends on whether abstracts are included in your conception of the physical world.

Quote:So from Hilbert's Hote we get:
infinity + infinity = infinity
infinity + infinity = infinity/2
infinity - 1 = infinity
infinity / 2 = infinity
infinity - infinity = 3

These are contradictory statements resulting from simple arithmetic operations (from 2). 

You CANNOT use infinite set theory constraints to explain away the contradictions because infinite set theory is not derived from a logical process. It is assumed in mathematics by the Axiom of Infinity. So the defeater you offer is exactly the same as "let's assume there is no contradictions". 

The funny thing is you don't need to be familiar with the nuances of set theory to see how there are no logical contradictions. Just think logically, if not mathematically.

So if you can't handle the mathematics. Let me help you with some simple logic.

The same collection of infinite things subtracted from the exact same collection of things is going to be equal to 0 (empty set/collection). Yes or no?

When you have a collection of infinite number of, say, apples ... and then you add to the collection an infinite number of oranges ... you get an infinite number of both apples and oranges, which now happens to be a different collection from the individual collection of apples and the individual collection of oranges. Yes or no?

Now, if you were to remove the infinite collection of oranges only from the total infinite collection of oranges and apples, you still end up with the infinite collection of apples. Yes or no?

If, instead, you remove ALL the apples and oranges from the total collection, except for 2 apples and 1 orange, then you logically end up with just 3 fruits. Yes or no?

What is finity - finity? What is finity + finity? In terms of numbers, the answer is not clear without context. Does this mean finity is not logically possible?

You really need to make an effort to address what I'm actually saying if you want to have a chance at stumping me. Can you please, for once, address this part directly?

Quote:LOL. read it again or ask Poly to clarify. You would get a kudo from those people if I said "white" and you said "black".

Read it again. There was a slight misunderstanding of my position, that's all. A position which, by the way, is Sean Carroll's position as well. And multiple other physicists. The only difference being semantic, in that I use "illusion" when Sean Carroll uses "derived" instead. Either way, causality is not a fundamental aspect of reality.

I really don't think you understood what that brief exchange was about. This is not my failure, but yours.

(March 2, 2018 at 2:18 pm)RoadRunner79 Wrote:

(March 2, 2018 at 12:42 pm)pocaracas Wrote: Here we should introduce the concept of the infinitesimal. Let's call it dx. dx is simply the limit, as x goes to zero, of x, or, in notation: lim(x->0) x.

How would you describe the limit as X goes to zero?

I suspect that when you include this, we are not that far apart in our thinking about the topic.  That this allows you to end the infinite regression (or progression) and reach the destination.

The limit as x goes to zero only exists if two other limits exist: Limit as x goes to 0+ and limit as x goes to 0-; these refer to the side from which zero is approached.
If these two limits are the same, the function is said to be continuous. If they are not, there is a discontinuity.
If both limits are +∞ and/or -∞, then you have a vertical asymptote.

Much like the concept of infinity (∞), the infinitesimal dx is also just a conceptual aid, it's not a number.

(March 1, 2018 at 12:04 pm)SteveII Wrote:

(March 1, 2018 at 10:08 am)Grandizer Wrote: What do you mean?

Don't you agree that 0.999999... actually is equal to 1? And not just potentially?

That's ridiculous. 0.99999... will NEVER equal 1. Not even potentially

In no world does that make any sense to assert that.

You're objectively wrong.

It's the same fucking number written differently.

---

(March 1, 2018 at 12:09 pm)Grandizer Wrote:

(March 1, 2018 at 12:04 pm)SteveII Wrote: That's ridiculous. 0.99999... will NEVER equal 1. Not even potentially

In no world does that make any sense to assert that.

---

(March 1, 2018 at 12:06 pm)pocaracas Wrote: dude...
1/3 = 0.3333333....
2/3 = 0.6666666....
3/3 = 0.9999999.... = 1

Poor Steve, and just to spoonfeed/confuse him a little more:

1/3 * 3 = 1
0.333333... * 3 = 0.999999...

1/3 = 0.333333...

1/3 * 3 = 0.333333... * 3 = 1

1/3 * 3 = 0.999999... = 1

And to put it logically.... there is no possible space between 0.999999... and 1 to make it a separate number. It's no more or less than 1, ergo it's 1.

(March 3, 2018 at 11:34 am)pocaracas Wrote:

(March 2, 2018 at 2:18 pm)RoadRunner79 Wrote: How would you describe the limit as X goes to zero?

I suspect that when you include this, we are not that far apart in our thinking about the topic.  That this allows you to end the infinite regression (or progression) and reach the destination.

The limit as x goes to zero only exists if two other limits exist: Limit as x goes to 0+ and limit as x goes to 0-; these refer to the side from which zero is approached.
If these two limits are the same, the function is said to be continuous. If they are not, there is a discontinuity.
If both limits are +∞ and/or -∞, then you have a vertical asymptote.

Much like the concept of infinity (∞), the infinitesimal dx is also just a conceptual aid, it's not a number.

A couple of comments:

1. Infinitesimals are no longer used. They were problematic in several ways and were replaced by the epsilon-delta definition of limit, which does not require them.

2. For a function to be continuous, the limit from the left and the limit from the right have to be equal *and* that limit has to be the value of the function. The two one-sided limits being equal just means the limit exists. Continuity means that limit is also the value of the function (obtained by plugging in).