(February 13, 2018 at 5:39 pm)polymath257 Wrote: OK, first some definitions.

Finite means it can be counted by some counting number or some 'real' number. Infinite means it is not finite.

So, a line segment of length 1 inch is 'finite' in length, but has an infinite number of points on it, so is infinite in terms of number of components. It is 'bounded', but also infinite. Similarly, anything with finite, non-zero volume will have an infinite number of components.

(This goes to show the definition of infinite in terms of boundedness is not a very good one).

The negative integers {..,-3,-2,-1} are unbounded below and bounded above. This is an infinite set because the number of negative integers cannot be counted by a finite number. That doesn't detract from its being bounded above (by 0). This is an example of a set that is bounded in one sense, but unbounded in another.

So, yes, you can have an actual infinite (like the negative integers) that is also bounded above. Not finishing isn't a criterion: the set is complete in and of itself.

Since I don't believe in any God, I can't say what you think about his/her/its infinite nature.

The water example shows that large collections can appear to be continuous when in fact they are not. So?

I'll try to comment on the other stuff later.   However I've had the discussion concerning a infinite number of points on a finite line before here (and haven't gotten very good answers). 

How are you defining a point, and on what basis do you state that there are an infinite number of them on a 1 inch line?

Zeno's paradox is a classical example of the points between two boundary limits.  The notion that you can always divide the distance between points in half, declaring a new halfway point...and will never run out of space to make such divisions (though they may become exceedingly small).  The quip, ofc, is that Achilles can never catch up to the tortoise, and so concedes the race.

Using a small number, it's easy to see how that works.  Between 0 and 1 is .5.  0 and .5 .25.  0 and .25 .125.  0 and .125 .0625. .03125, .015625, .0078125. On and on, ad infinitum. Hilariously, if you add up all of these infinite and tiny distances you will only get back to a sum total of 1. Coincidentally, 1 meter is all the tortoise needed to swindle Achilles. Talked him right out of his racing boots.

There are always more points between any 2 points. It's a tenet of geometry.

(February 13, 2018 at 5:54 pm)RoadRunner79 Wrote:

(February 13, 2018 at 5:39 pm)polymath257 Wrote: OK, first some definitions.

Finite means it can be counted by some counting number or some 'real' number. Infinite means it is not finite.

So, a line segment of length 1 inch is 'finite' in length, but has an infinite number of points on it, so is infinite in terms of number of components. It is 'bounded', but also infinite. Similarly, anything with finite, non-zero volume will have an infinite number of components.

(This goes to show the definition of infinite in terms of boundedness is not a very good one).

The negative integers {..,-3,-2,-1} are unbounded below and bounded above. This is an infinite set because the number of negative integers cannot be counted by a finite number. That doesn't detract from its being bounded above (by 0). This is an example of a set that is bounded in one sense, but unbounded in another.

So, yes, you can have an actual infinite (like the negative integers) that is also bounded above. Not finishing isn't a criterion: the set is complete in and of itself.

Since I don't believe in any God, I can't say what you think about his/her/its infinite nature.

The water example shows that large collections can appear to be continuous when in fact they are not. So?

I'll try to comment on the other stuff later.   However I've had the discussion concerning a infinite number of points on a finite line before here (and haven't gotten very good answers). 

How are you defining a point, and on what basis do you state that there are an infinite number of them on a 1 inch line?

Between any two points, there is another that is halfway that is different than both ends. That alone is enough to guarantee an infinite number of points. And that is true no matter what the definition.

(February 12, 2018 at 8:28 am)polymath257 Wrote:

(February 12, 2018 at 7:36 am)Grandizer Wrote: But how does one even start the adding if there is no start to it? And how does the adding have already happened without a start?

Well, that is sort of the whole point. There *is* no start. It is an ongoing process with no start.

Again, think of the integers (.....,-4,-3,-2,-1,0,1,2,3,4,.....)

There is no start and there is no finish. Every integer has an immediate predesessor and an immediate successor.

And again, this is not a proof this happens in the real world. It is, however, a demonstration that there is no *logical* issue with an infinite regress.

I think this is a fine example.  Time just does seem like the kind of thing about which you can ask about the past moment or the nth moment before the current one, or, the next or nth moment after the current one.  Approached like this, there is no need to (or sense in) asking about any first moment or first number.  The limitless reach of each is accounted for without any puzzling first number or first moment.

(February 13, 2018 at 6:01 pm)Khemikal Wrote: Zeno's paradox is a classical example of the points between two boundary limits.  The notion that you can always divide the distance between points in half, declaring a new halfway point...and will never run out of space to make such divisions (though they may become exceedingly small).  The quip, ofc, is that Achilles can never catch up to the tortoise, and so concedes the race.

Using a small number, it's easy to see how that works.  Between 0 and 1 is .5.  0 and .5 .25.  0 and .25 .125.  0 and .125 .0625.   .03125, .015625,  .0078125.  On and on, ad infinitum.  Hilariously, if you add up all of these infinite and tiny distances you will only get back to a sum total of 1.  Coincidentally, 1 meter is all the tortoise needed to swindle Achilles.  Talked him right out of his racing boots.

And, just to point out the solution to Zeno's paradox: there are also an infinite number of subdivisions of time, so you can go through an infinite number of spatial points in a similarly infinite number of temporal points.

---

(February 13, 2018 at 5:54 pm)RoadRunner79 Wrote: I'll try to comment on the other stuff later.   However I've had the discussion concerning a infinite number of points on a finite line before here (and haven't gotten very good answers). 

How are you defining a point, and on what basis do you state that there are an infinite number of them on a 1 inch line?

Mathematically, the points on a line correspond to decimal numbers between 0 and 1 (the two ends of the one inch long line segment. There is an infinite number of such decimals.

More specifically, if you have a finite collection of points on that line segment, find two next to each other and you can find a point half way between them. That shows the finite list was not complete.

In other words, the collection of points is infinite.

Now, even more is true: there are different sizes of infinite set mathematically. The infinite set of counting numbers {1,2,3,4,5,...} turns out to have the smallest infinite 'size' (technically, cardinality). It turns out that the collection of ALL integers (positive and negative) has that same 'size' because there is a way to pair off al positive counting numbers and all integers.

1<->0
2<->1
3<->-1
4<->2
5<->-2
6<->3
7<->-3
.
.
.

It turns out that the collection of all decimal numbers between 0 and 1 is a *larger* infinity than the infinity for counting numbers.

(February 13, 2018 at 6:08 pm)polymath257 Wrote:

(February 13, 2018 at 5:54 pm)RoadRunner79 Wrote: I'll try to comment on the other stuff later.   However I've had the discussion concerning a infinite number of points on a finite line before here (and haven't gotten very good answers). 

How are you defining a point, and on what basis do you state that there are an infinite number of them on a 1 inch line?

Between any two points, there is another that is halfway that is different than both ends. That alone is enough to guarantee an infinite number of points. And that is true no matter what the definition.

Yes, that is the assertion, and it sounds good.   However if you are saying that there is an infinite number of something, then I want to know an infinite number of what (definition) and how did you determine that they where infinite?     I'm just guessing, but I don't think that you counted them.   And you can check my math, but I think that dividing by two will always return a finite number.

(February 13, 2018 at 6:21 pm)RoadRunner79 Wrote:

(February 13, 2018 at 6:08 pm)polymath257 Wrote: Between any two points, there is another that is halfway that is different than both ends. That alone is enough to guarantee an infinite number of points. And that is true no matter what the definition.

Yes, that is the assertion, and it sounds good.   However if you are saying that there is an infinite number of something, then I want to know an infinite number of what (definition) and how did you determine that they where infinite?     I'm just guessing, but I don't think that you counted them.   And you can check my math, but I think that dividing by two will always return a finite number.

Yes, an infinite number of finite numbers.

I gave the argument above for the collection being infinite. It is a simple argument. As long as you admit that there is a point between any two given ones (no matter what the definition), the result follows.

(February 13, 2018 at 6:25 pm)polymath257 Wrote:

(February 13, 2018 at 6:21 pm)RoadRunner79 Wrote: Yes, that is the assertion, and it sounds good.   However if you are saying that there is an infinite number of something, then I want to know an infinite number of what (definition) and how did you determine that they where infinite?     I'm just guessing, but I don't think that you counted them.   And you can check my math, but I think that dividing by two will always return a finite number.

Yes, an infinite number of finite numbers.

I gave the argument above for the collection being infinite. It is a simple argument. As long as you admit that there is a point between any two given ones (no matter what the definition), the result follows.

I think that there a couple of problems with this as a refutation.   I think that you point is an abstraction, which is why you do not want to define it.  I also think that if you do define it, as anything physical and finite, your argument fails.  It's also on this point, that you have an issue, because the contention is that you cannot have an infinite number of things by successful addition (or division or multiplication for that matter).   No one is arguing that you cannot have an infinite set of abstract numbers (which is really all I think your points are).  If you disagree, then give a definition which is not an abstract, and we can test it.  I think that your point is zero (or nothing), or always changing and not talking about the same thing.

Secondarily similar to successive addition, you will never reach an actual infinite by successive division.  At any given step in the process, you only have a potential infinite.  You don't posses and actual infinite at any time, but are extrapolating out the action infinitely.

...and now we're arguing against simple arithmetic.

Good lord..why!?! Oh yeah, that's right, because somebody's silly idea of god was tied up with being wrong about math. Meh, wouldn't be the first time. 1+1+1 isn't 1, either.