Thinking about infinity

[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 wish I could correct your ignorance [edit] but I don't think I can. Do you think it is impossible for an actual infinity to exist, and if so, what reasons would you give?

It's inconceivable.

[SteveII]

My understanding is that infinities are useful fictions in math equations. When used in physics, they are always converted back to finite quantities 

So actual infinities do not exist in reality? Do you think it is even possible for an infinite set of things to exist as a single thing?

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

[Ignorant]

...As to how a finite object can be made up of infinitely many parts, that is covered all the time (at least abstractly) in mathematics by such processes as summing an infinite geometric series, or performing integration to calculate the area under a curve...

So, should we consider an infinite geometric series (and/or its sum), or integration as real "only in the abstract" or "a step removed from reality"?

[Ignorant]

I wish I could correct your ignorance [edit] but I don't think I can. Do you think it is impossible for an actual infinity to exist, and if so, what reasons would you give?

It's inconceivable.

Does inconceivability suggest logical impossibility?

[Ignorant]

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?

[robvalue]

...As to how a finite object can be made up of infinitely many parts, that is covered all the time (at least abstractly) in mathematics by such processes as summing an infinite geometric series, or performing integration to calculate the area under a curve...

So, should we consider an infinite geometric series (and/or its sum), or integration as real "only in the abstract" or "a step removed from reality"?

Well, it's logically possible that it could map accurately to reality. So we know it is possible abstractly, but it remains only a logical possibility for reality.

We have made a model of a real object, and our results may or may not accurately map back to reality. This is where evidence and testing comes in. Our resident QM experts may well have a decent opinion regarding this. I am not qualified to say. But ultimately, reality holds the truth and we're just trying to get close to it.

This discussion is about the extreme limits of reality, and as such our models are going to be increasingly difficult to test. Normally we don't require such an extreme level of accuracy.

[Ignorant]

To review what I said about dividing something up infinitely... that too would appear to be a logical possibility. But this is something we would potentially have more information on, so those well versed in QM may have a reasonable prediction as to whether it is possible based on what we observe. (Something about "stable wave formations" comes to mind.) However, there's always the chance that this prediction is wrong, and that we're just using the wrong techniques or tools. Certainty is not the job of science.

Yes but in what sense? A line segment can be divided infinitely, sure. Does that make a line segment a finite infinity of things?

[robvalue]

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?

[TheRealJoeFish]

So actual infinities do not exist in reality? Do you think it is even possible for an infinite set of things to exist as a single thing?

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

---

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

[Alex K]

yeah... e.g. by taking power sets. A very fascinating construction.