Thinking about infinity

[Ignorant]

But, of course, there are different levels of infinity as well.  For instance, you can famously prove that there are the "same" amount of natural numbers and rational numbers, but there are "more" real numbers than there are natural numbers.

I remember in Real Analysis 2 we discussed Omega (the "level" of infinity of the reals), which is greater than Aleph_0 (the level of infinity of the naturals), and then how you could make sets of things with different measures of infinity, like Omega^2 and such

Do you think that tells us anything about the logical possibility of an infinite set which has every member simultaneously existing in reality?

[SteveII]

I think that the set of real numbers is infinite. That is abstract though. 1...2...3...4 are real numbers. 

Have you ever read about Hilbert's Hotel:

Imagine a hotel with a finite number of rooms. All the rooms are full and a new guest walks in and wants a room. The desk clerk says no rooms are available.

Now imagine a hotel that has an infinite number of rooms. All the rooms are filled up so an infinite number of guests. A new guest walks up and wants a room. All the clerk has to to do is to move the guest in room #1 to room #2 and the guest from #2 to #3 and so on so your new guest can have a room #1. You can do this infinite number of times to a hotel that was already full.

Now imagine instead the clerk moves the guest from #1 to #2 and from #2 to #4 and from #3 to #6 (each being moved to a room number twice the original). All the odd number rooms become vacant. You can add an infinite number of new guests to a hotel that was full and end up with it half empty. 

How many people would be in the hotel if the guest in #1 checked out?

If everyone in odd number rooms checks out, how many checked out? How many are left?

Now what if all the guest above room number 3 check out. How many checked out? How many are left?

So from the above we get:
infinity + infinity = infinity
infinity + infinity = infinity/2
infinity - 1 = infinity
infinity / 2 = infinity
infinity - infinity = 3

I have read about this, and I'm glad you brought it up. So is it logically possible to fill every room of a hotel with an infinity of rooms?

No. It is illustrating that a real infinite quantity is not coherent.

[RoadRunner79]

Some sets of infinities certainly seem logically possible. 

For example, the infinite set of natural real numbers seems to exist, which is to say that there is an infinity of natural real numbers existing even now, and there is none of them which do not yet exist.

There are an infinity of points on a line segment.

It is logically possible that reality has an infinite history of cause and effect (i.e. extending "backward" through the big-bang), and that it will have an infinite future.

In what sense could an infinity of simultaneously existing things exist together as a finite thing?

The set of natural real numbers aren't really infinite, we only think of them as such. Same holds for the points on a line segment. In the real world, nothing has been proven to be infinite, nor could it be. Feel free to dream of it though.

In my uninformed, uneducated opinion infinity is illogical and doesn't have a place in the real world.

Do correct my ignorance.

I am confused by your statement, about the abstract of real numbers, not being infinite. Could you clarify? I do agree about the line though.... In any case you are going to have a finite number of points. You can redefine what the point means, but then you still have a finite number.

[TheRealJoeFish]

First, it may be true that there can only exist a "finite number of things" (whatever that means) in a universe, but maybe for a different reason than the obvious one: there may be a lower limit on the "size" of something, and thus a limit on the number of things that can be in a given place. Expressed a different way - and, as always when I'm talking about physics, I ask Alex K to elaborate if what I'm saying makes sense or correct me if it doesn't - if there's a finite amount of energy in the universe, and a lower limit on the smallest "amount" of energy that can be transferred at a given time, that may necessarily constrain the number of "things" that can exist. I'm thinking Planck lengths and scales and such, based on my rudimentary understanding of quantum physics mumbo jumbo.

From my (far more rigorously cultivated) mathematical perspective, I think it's important when discussing "infinity in the real world" to distinguish between what I'll call "actual infinity" and "arbitrary largeness." For example, consider the coastline problem. If you try to measure the coastline of an island with a yardstick, you'll get a certain answer for the length of the coastline. Because the yardstick doesn't bend, though, you'll be missing lots of tiny ins and outs that are less than a yard in size. So, when you measure the coastline with a ruler, you'll get a bigger number. If you measure the coastline with rope, you'll get a bigger number still, because you'll be lining up the bendy rope with the edge of the water, but you'll still miss all of the little wiggles in the coastline that are smaller than the width of the rope. You'll get a bigger size if you measure with string, and bigger still if you measure with fishing wire (but you'll still be missing the myriad of microscopic wiggles smaller than the width of the fishing line, such as the contours of grains of sand). What this means is, you'll never measure the coastline to be "infinity", but, if you pick any length, there's a level of precision at which the measurement of the coastline will be greater than that length.

Or, maybe a better example, with the line: Sure, you may never be able to count an "infinite" number of points on a line. But, if I give you any number at all (like, 52 billion), with enough time you could count more points on the line than that number.

[Excited Penguin]

The set of natural real numbers aren't really infinite, we only think of them as such. Same holds for the points on a line segment. In the real world, nothing has been proven to be infinite, nor could it be. Feel free to dream of it though.

In my uninformed, uneducated opinion infinity is illogical and doesn't have a place in the real world.

Do correct my ignorance.

I am confused by your statement, about the abstract of real numbers, not being infinite. Could you clarify?   I do agree about the line though....  In any case you are going to have a finite number of points. You can redefine what the point means, but then you still have a finite number.

How exactly do you think that real numbers are infinite?

[Ignorant]

...  Expressed a different way - and, as always when I'm talking about physics, I ask Alex K to elaborate if what I'm saying makes sense or correct me if it doesn't - if there's a finite amount of energy in the universe, and a lower limit on the smallest "amount" of energy that can be transferred at a given time, that may necessarily constrain the number of "things" that can exist...

...  Consider the coastline problem.  ...  What this means is, you'll never measure the coastline to be "infinity", but, if you pick any length, there's a level of precision at which the measurement of the coastline will be greater than that length...

...Sure, you may never be able to count an "infinite" number of points on a line.  But, if I give you any number at all (like, 52 billion), with enough time you could count more points on the line than that number.

This is very helpful, and I do hope Alex K can help out at his convenience. I see a few different aspects in play here. Please correct any of the following:

1) The current models of the universe [may?] exclude the possibility of an actual infinity of things, but as models, these are subject to being replaced by better models and new evidence.

2) I found the coastline problem very helpful. Even though the accuracy with which you can measure the coastline's length can be improved ad infinitum, the length itself being measured is approached (and perhaps never reached) as a finite boundary. The process of increasing accuracy may be considered notionally infinite (like a geometric series?), while the coastline's length remains finite.

3) The infinite divisibility of a line is similar to the coastline problem. The "process" of identifying (counting) more and more points on that line may proceed ad infinitum. The infinity of points on a line are notionally infinite, rather than an infinity of points all simultaneously being the length of the line? <= This one is still a bit fuzzy.

[Ignorant]

I have read about this, and I'm glad you brought it up. So is it logically possible to fill every room of a hotel with an infinity of rooms?

No. It is illustrating that a real infinite quantity is not coherent.

Would you mind elaborating?

[Ignorant]

Yes, if it could be divided up infinitely in reality, it would be a finitely "long" object, which could be made up of infinitely many parts of increasingly small lengths.

Of course, where you make the cuts is arbitrary, and it remains the sum of those parts whether or not you actually cut it. It's another way of viewing the same object.

In the same way: when you walk a metre, how many times have you passed the halfway point, if we recalculate it every time we reach that point based on the remaining length?

Right, I think this is one of Zeno's paradox's (or at least similar to it). I think it serves as a good example. The process you describe can proceed with an infinity of recalculations. So here's a few questions to consider:

Does/Can the infinity of  "halfway points" actually exist as a reality on the length of one meter (somehow simultaneously "making up" the length of the meter, or do they exist notionally as a geometric series represents one which can always be added to, or like the coastline problem explained above)?

Consider a "reversal" of the recalculation. Could you ever "begin" to walk at all if your first distance you must pass is the infinitely smallest halfway point? <= Is that even logically coherent?

[robvalue]

Yes, you're correct, it can be to do with Zeno's paradox.

It's the problem of insisting we only take finitely many steps and thus never reach our target. But (abstractly at least) we take infinitely many "steps"; each of smaller and smaller lengths.

In reality, it may or may not work like this. Is there a minimum distance something can move? If there is, then you can't take infinitely many steps. There is only a finite number you can take, no matter how small, between two points a finite distance apart. Your first step must be at least this amount (it would likely we would be dealing with multiples of this smallest amount I think). We would in effect be "jumping" from one point to another, without ever occupying the space inbetween.

But if there is no minimum amount, then you have moved through infinitely many. Disallowing this would, as you correctly observe, prevent you from even starting to move at all. But you are essentially stopping the "time taken" from ever reaching a certain point, which is not valid.

It's a weird concept to be sure. But there is no problem at all with a finite number being an infinite sum, abstractly. The only question is whether reality can be subdivided the same way. If there is no limit to how small I can cut something up, I could go on forever. Obviously, if I'm only allowed to make finitely many cuts, I can't make them all. But that's the essence of Zeno's paradox. Insisting on finitely many is the equivalent of insisting I cannot move in the first place. It's assuming the conclusion by disallowing infinities as a premise.

So, for example, let's say an object is of length 1 metre. Let's say there is no minimum length that a subdivision can be. Then, if we ask how many subdivisions of it are there if we use these lengths (I used this example before):

1/2 + 1/4 + 1/8 + ...

then the answer is infinitely many. It doesn't matter if I actually go in and mark them all, obviously that's impossible in a finite amount of time if it takes me a fixed amount of time for each cut. But if the time taken is proportional to the distance I have to move instead to make the cut, then I can theoretically make all the cuts.

It does consist of an infinite amount of them, whether or not they are actually "marked" by someone, or cut into pieces. However, if we reach a minimum distance we can cut, this no longer works. Were confined to (length/minimum distance) numbers of partitions.

Let X = 1/2 + 1/4 + 1/8 + ...

Then 2X = 1 + 1/2 + 1/4 + 1/8 + ...

2X - X = 1 [the rest of the terms all cancel out]

X = 1

This isn't a trick, it's a genuine mathematical method.

[Ignorant]

Yes, you're correct, it can be to do with Zeno's paradox.

It's the problem of insisting we only take finitely many steps and thus never reach our target. But (abstractly at least) we take infinitely many "steps"; each of smaller and smaller lengths.

In reality, it may or may not work like this. Is there a minimum distance something can move? If there is, then you can't take infinitely many steps. There is only a finite number you can take, no matter how small, between two points a finite distance apart. Your first step must be at least this amount (it would likely we would be dealing with multiples of this smallest amount I think). We would in effect be "jumping" from one point to another, without ever occupying the space inbetween.

But if there is no minimum amount, then you have moved through infinitely many. Disallowing this would, as you correctly observe, prevent you from even starting to move at all. But you are essentially stopping the "time taken" from ever reaching a certain point, which is not valid.

It's a weird concept to be sure. But there is no problem at all with a finite number being an infinite sum, abstractly. The only question is whether reality can be subdivided the same way. If there is no limit to how small I can cut something up, I could go on forever. Obviously, if I'm only allowed to make finitely many cuts, I can't make them all. But that's the essence of Zeno's paradox. Insisting on finitely many is the equivalent of insisting I cannot move in the first place. It's assuming the conclusion by disallowing infinities as a premise.

So, for example, let's say an object is of length 1 metre. Let's say there is no minimum length that a subdivision can be. Then, if we ask how many subdivisions of it are there if we use these lengths (I used this example before):

1/2 + 1/4 + 1/8 + ...

then the answer is infinitely many. It doesn't matter if I actually go in and mark them all, obviously that's impossible in a finite amount of time if it takes me a fixed amount of time for each cut. But if the time taken is proportional to the distance I have to move instead to make the cut, then I can theoretically make all the cuts.

It does consist of an infinite amount of them, whether or not they are actually "marked" by someone, or cut into pieces. However, if we reach a minimum distance we can cut, this no longer works. Were confined to (length/minimum distance) numbers of partitions.

Let X = 1/2 + 1/4 + 1/8 + ...

Then 2X = 1 + 1/2 + 1/4 + 1/8 + ...

2X - X = 1  [the rest of the terms all cancel out]

X = 1

This isn't a trick, it's a genuine mathematical method.

I certainly agree with all of that. Now Consider X = 1/(2^n). The sum of X, for n beginning at 1 and approaching infinity is coherent. The series begins with a known finite value, and adds its half, and adds the half of that, ad infinitum. That makes perfect sense.

How would you represent the reverse? In other words, let n = ∞ and then approach 1. <= What is this like?