Actual Infinity in Reality?

[RoadRunner79]

This is certainly an interesting conversation. In the interest to advance it further, I offer the following observations about some of the discussion so far.

It seems to be repeated often, that many cosmologist do not have  an issue with an actual infinity or an infinite universe.  However, whether this notion is popular or not, what matters is the reason behind their opinions.  From what I understand, I do not believe that they are referring to some expertise in their field, nor to some understanding or observation of the universe, which is little known to the lay person.  Even if granted for the sake of argument, In my opinion; I doubt that it is even possible to measure something as being infinite.  About the most that we gather from science is a nearly flat curvature of the universe, which could be consistent with, but does not necessarily require an infinite universe.

The question has been asked, if the universe is not infinite, then what is space expanding into.  I think that this is a good question, and I do not know the answer.   But it seems that it poses more of an issue if space is already and entirely completed infinite.  The question still remains, and I don't see why an answer for one, would not work in the other.  Accept that with an infinite, you already have the universe being limitless.

It is argued, that math has axioms, that deal with infinite sets.  Ok.... I don't see where anything which has been presented deals with the topic of an actual vs a potential infinite.  I also do not believe that the extrapolations or concept of infinity count as an actual or completed infinite.  Not in the sense of the philosophical objections.   Perhaps it fits the definition of a completed set, but I don't think that this is meeting the philosophical objections.

I do agree, and would say that I am more aware of the arguments that infinity isn't a number, and therefore shouldn't be treated like a number.  Thanks to those who brought this to attention.    Infinity means endless or without limit.  I think it's also very easy, as I catch myself and others who would say it is not a number, can easily slip back into that bad pattern of thought.

For Zeno's paradox,  it was offered that calculus is the answer, and that a infinite small amount of time, and an infinitely small distance can be covered.  I don't buy it.   It doesn't matter if you have a lot of time, or infinitely small amounts of time, or however much time.  If you follow that procedure of infinitely dividing in half the remainder, you will never reach the destination.  It's not a matter of time.   This may also be demonstrating the issue above of using infinity as a number.  Also, you cannot have an infinite number of small distances in a finite distance, nor an infinite amount of time, in a finite time.  This is a contradiction.  Zeno in his paradox, was arguing that there was no motion.  And we can easily observe that his conclusion is false.  Not only can I set out to and reach a distance of 10 ft.  I may very well pass it.  Who are you going to believe "Zeno... or you lying eyes?"  There is nothing wrong with Zeno's math, the simplest answer is that there is a problem with his underlying assumptions.  

If I'm understanding correctly, I don't think that anyone is making an argument for an actual infinity, only that it is possible.  And if scientist are not saying much about from their field about actual infinities, then I don't see why the are being brought up.   Mathematicians have assumptions about infinities, but are not working towards demonstrating an actual infinity, but assuming that at least an abstract set of infinite numbers exist.  Which is how the system is set up to begin with, and you can always add, and carry over the remainder (potentially endlessly).  But there isn't a set of numbers floating out there in the either.  Even our finite minds cannot hold a endless amount of numbers.  It can only try to grasp the concept.  And this is where I think that the contradiction lies.  In an infinite number of things is complete and an actual infinite is to say that it exists in it's entirety.  However it's endless you can always add one more, that wasn't there before.  If you cannot add one more, then it is not infinite.

But these are only two tools to truth.  Philosophers have had objections to infinity for a long time.  They have paradoxes.  And I believe contradictions.  Something can not be infinite and finite at the same time.  It cannot be endless and have an end.  It cannot be completed and incomplete a moment later.  And even if you are not convinced that these are contradictions (infinity can be a difficult thing to grasp and you or I may be in error)  There are still the paradoxes or absurdities.  No matter how you slice it, it seems that there is a lot more reason to doubt an actual infinity and I see no reasons to believe for it.

[GrandizerII]

Philosophers have had objections to every single argument one could think of. There is no consensus among philosophers regarding actual infinity. AFAIK, at least. Unless we zoom in on philosophers of mathematics only, in which case, probably there is in favor of actual infinity being logically/metaphysically possible (but not certain).

[RoadRunner79]

Philosophers have had objections to every single argument one could think of. There is no consensus among philosophers regarding actual infinity.

I agree about their not being a consensus.... but we all have a brain and can reason (though some may be more disciplined than others).  We can think for ourselves and look at the arguments and objections that others have made.    We may disagree, but in the end, I want to at least say that I thought about it, and I evaluated and answered others objections.

[GrandizerII]

We may disagree, but in the end, I want to at least say that I thought about it, and I evaluated and answered others objections.

Perhaps not enough to readjust your views though, right?

I'm guessing based on this quote:

In an infinite number of things is complete and an actual infinite is to say that it exists in it's entirety. However it's endless you can always add one more, that wasn't there before. If you cannot add one more, then it is not infinite.

The answer is you haven't changed your mind on anything to do with actual infinity. You clearly just can't accept that all the elements are already there in such a collection, so there's nothing more of the same type of element to add.

[RoadRunner79]

We may disagree, but in the end, I want to at least say that I thought about it, and I evaluated and answered others objections.

Perhaps not enough to readjust your views though, right?

I'm guessing based on this quote:

In an infinite number of things is complete and an actual infinite is to say that it exists in it's entirety. However it's endless you can always add one more, that wasn't there before. If you cannot add one more, then it is not infinite.

The answer is you haven't changed your mind on anything to do with actual infinity. You clearly just can't accept that all the elements are already there in such a collection, so there's nothing more of the same type of element to add.

So, you are at an end with nothing more to add?

[GrandizerII]

Perhaps not enough to readjust your views though, right?

I'm guessing based on this quote:


The answer is you haven't changed your mind on anything to do with actual infinity. You clearly just can't accept that all the elements are already there in such a collection, so there's nothing more of the same type of element to add.

So, you are at an end with nothing more to add?

There is no end, but no matter how far you go through the set, there is already an element to observe.

---

So, you are at an end with nothing more to add?

There is no end, but no matter how far you go through the set, there is already an element to observe.

Just use your imagination and assume all elements actually already exist in the set of positive integers:

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...}

The elements are all already there, including 11 and so on. Can you add more positive integers to the set that aren't already there yet? No! No matter how far you go through the set, any last integer you reach would have already been an element of the set. It didn't need your observation to bring it into being. Even if it was Graham's number, or TREE(3), they're already there (assuming they're integers, of course; if not, ignore this last sentence).

[polymath257]

I want to point out that even asking these questions assumes the set theory you reject. When you talk about the infinite number of rooms of the HH, you are using the methods of set theory.

1. That's ridiculous. HH posits an infinite amount of real objects and then illustrate how it behaves absurdly when you attempt to make changes. Clear proof that an actual infinity contains absurdities that cannot be reconciled without applying a series of assumptions to make sense of it. It is in trying to make sense of it that YOU keep bringing in set theory. 

1. Not quite. The notion of 'half' of the rooms is not well defined in this context. There is a notion of density for subsets of the natural numbers, and that density is 1/2, but that is a different thing *and relies on the set theory you object to*. The notion of 'half' does not apply here unless you define what it means in this context. And *that* requires the set theory you reject.

2. An infinite number. More specifically, *in the set theory your reject* the answer is aleph_0.

3. The answer to both is 'an infinite number'. More specifically, *in the set theory you reject*, the answers are both aleph_0.

4. The answer is 3. We are using set theoretic differences here, though, *which you reject*.

5. This is correct, although you haven't defined what addition is for this context and cannot do so without the set theory you reject. But, yes, the union of two disjoint infinite sets is infinite *in the set theory you reject*.

6. As described above, the notion of infinity/2 is not well defined, although *with the set theory you reject* it can be given a meaning and the result is again aleph_0.

7. Again, your demonstration is that when you remove an element from an infinite set (which is a process you reject), you get another infinite set of the same cardinality. That is true.

8. As explained above, infinity/2 is meaningless unless you define your operations. But yes, if you take an infinite set and divide it into two sets of the same cardinalities, then both pieces will have the same cardinality as the original.

9. And once again, you are attempting to generalize from the case of finite sets to that of infinite sets. The notion of subtraction of cardinalities is not well defined. All this means in this context is that if you take an infinite collection and remove an infinite collection, the answer will depend on *how* you remove that infinite collection. That isn't a contradiction.

2. I think you literally invoked set theory in every reply. So again you use a mathematical assumption to explain why we get 5 absurd results when moving people around. That is question begging and is insufficient as an explanation let along a defense of an actual infinity. 

No, what you have shown is that the operations of division and subtraction of infinites are either not well defined (they depend on specifics of the situation) or they give results you see as paradoxical, even though they are consistent. But guess what? That happens in ordinary arithmetic. What is 3*0? What is 5*0? So is 0/0=3 or is 0/0=5? That we cannot divide by 0 doesn't mean it is contradictory to use the number 0.

3. No, what HH shows is that the concept of infinity is not equipped to explain real objects. The conclusions are true, yet they are absurd. Regarding "0", you cannot parody HH with 0 and show anything remotely similar. As such, the comparison you are attempting makes no progress toward "normalizing" the absurd conclusions. 

So, here is a question: why do you expect the operations of subtraction and division would work the same way for infinite things as for finite things? What is the physical basis for this expectation?

4. For the very obvious reason that you need to be able to subtract and divide real objections in the real world. If you can't do that with infinite quantities, you cannot have an actual infinity of real objects. This is not rocket science. For like the ninth time, don't you find it interesting that you cannot find an article defending an actual infinity of real objects to post for us to prove your point? 

---

Let me ask this: do you think all those things make sense *within* the context of an infinite set theory?

5. Real objects MUST be able to be divided and subtracted because they are tangible things and you HAVE TO be able to do these simple operations with tangible things. 

---

 Consciousness is basic for logic? Really?

OK, what are LNC and LEM? Those are abbreviations I am not familiar with.

I would notice that you used the operations of addition, subtraction, and division. Those are not on your list, are they?

Yes a consciousness is an awareness and ability to perceive reality and therefore necessary in any discussion of reality. 

6. Law of Non-contradiction and the Law of the Excluded Middle. 

7. Operations of addition, subtraction and division are results of real objects being moved about. They are not axiomatic. I don't have to assume any additional truths before conducting the operations.

 6. Fair enough. Not an abbreviation I typically use.

7. OK, you can add two infinite collections together and get another infinite collection. No problem.

If you have an infinite collection and remove a part of it, the result you get depends on which part you remove. There is no requirement that the *abstract* notions of subtraction extend to infinite quantities. The operation of addition does (even though that isn't a logical requirement either). The operation of division isn't even well defined for integers, so your claim it is required for infinite quantities is silly.

5. OK, but the fact that you can do these operations on objects doesn't mean you have to get the same result when you do the operation in a different way. And that is precisely what you were doing.

4. For the ninth time, I don't expect to because any physicist that works on this sees infinite space, for example, as a logical possibility that will need to be answered via observation, not through metaphysics. The problem with subtraction and division is that *how* you do them affects the answer. if you keep track of that, there is no contradiction. Different operations will yield different results.

3. They are only absurd because you expect infinite quatitiplay by the same rules as finite ones. That is your problem, not the results.

2. NONE of the results were absurd. They may stretch your intuitions a bit, but there really isn't a problem with any of the results that are well defined (the ones involving addition, subtraction of a finite quantity, and division by a finite quantity). Those that are NOT well defined are still consistent *if* you go to the collections themselves and don't expect a 'numerical' answer in all cases.

The *exact* same thing happens when attempting to divide by 0. You can claim that we always need to divide, but this is a good example where it is perfectly reasonable to NOT allow division. Why? Because it would potentially give more than one answer. Since 3*0=0, we expect 3=0/0. Since 5*0=0, we expect 5=0/0. Because of this, we simply do not define 0/0. It is an operation we leave undefined. The same happens, for infinity-infinity and infinity/infinity. This is *completely* analogous.

1. Every single one of those operations gave an answer, didn't they? it may not have been the answer you wanted, but an answer was given.

If you take out every odd number, you have removed infinitely many things and have left infinitely many things. If you remove all but the first 3 numbers, however (a different operation), you will have removed infinitely many things from infinite many things and have finitely many left.  Where is the absurdity? Both operations gave well defined answers. The answers were different because the specific operations were different. That's all.

I bring in set theory because you have postulated an infinite collection of rooms. That collection of rooms *is a set*. That's all that sets are: collections of objects. In particular, the collection of rooms in the HH is an infinite set. Because of this, already adopted the axiom of infinity when you start asking these questions. So, it was not I that begged the question, but you.

At no point did you show a contradiction. You did show that certain operations are not well defined, but gave no reason to think they must be. But, at the same time, you gave the correct answers for every particular. Funny how you could do that if it is so unreasonable.

[GrandizerII]

Also, let me address this one as well:

For Zeno's paradox,  it was offered that calculus is the answer, and that a infinite small amount of time, and an infinitely small distance can be covered.  I don't buy it.   It doesn't matter if you have a lot of time, or infinitely small amounts of time, or however much time.  If you follow that procedure of infinitely dividing in half the remainder, you will never reach the destination.  It's not a matter of time.   This may also be demonstrating the issue above of using infinity as a number.  Also, you cannot have an infinite number of small distances in a finite distance, nor an infinite amount of time, in a finite time.  This is a contradiction.  Zeno in his paradox, was arguing that there was no motion.  And we can easily observe that his conclusion is false.  Not only can I set out to and reach a distance of 10 ft.  I may very well pass it.  Who are you going to believe "Zeno... or you lying eyes?"  There is nothing wrong with Zeno's math, the simplest answer is that there is a problem with his underlying assumptions.

Note that Zeno's paradox isn't an argument against a finite distance having an infinite number of smaller distances, or a finite time having an infinite number of smaller time instants. The argument is supposed to be against the possibility of traversing such a distance or time. And for that traversing bit, I leave it up to others to address for you.

[RoadRunner79]

So, you are at an end with nothing more to add?

There is no end, but no matter how far you go through the set, there is already an element to observe.

---

There is no end, but no matter how far you go through the set, there is already an element to observe.

Just use your imagination and assume all elements actually already exist in the set of positive integers:

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ...}

The elements are all already there, including 11 and so on. Can you add more positive integers to the set that aren't already there yet? No! No matter how far you go through the set, any last integer you reach would have already been an element of the set. It didn't need your observation to bring it into being. Even if it was Graham's number, or TREE(3), they're already there (assuming they're integers, of course; if not, ignore this last sentence).

What are you saying that the term “actual” means in the term.
I take it to mean that it is completed or actualized. Yet the term infinite means without limit or end... or in other words never completed. It is similar to a square circle.
There also would not be a total quantity of the set.

[polymath257]

For Zeno's paradox,  it was offered that calculus is the answer, and that a infinite small amount of time, and an infinitely small distance can be covered.  I don't buy it.   It doesn't matter if you have a lot of time, or infinitely small amounts of time, or however much time.  If you follow that procedure of infinitely dividing in half the remainder, you will never reach the destination.  It's not a matter of time.   This may also be demonstrating the issue above of using infinity as a number.  Also, you cannot have an infinite number of small distances in a finite distance, nor an infinite amount of time, in a finite time.  This is a contradiction.  Zeno in his paradox, was arguing that there was no motion.  And we can easily observe that his conclusion is false.  Not only can I set out to and reach a distance of 10 ft.  I may very well pass it.  Who are you going to believe "Zeno... or you lying eyes?"  There is nothing wrong with Zeno's math, the simplest answer is that there is a problem with his underlying assumptions. 

Let me give a brief treatment of Zeno's paradox.

Let's start with the one that says to move from point A to point B, you have to first reach the half-way point, then the half-way point to that, then the half-way point to that, etc.

The claim is that we cannot move through an infinite collection of points in a finite amount of time. So, what *actually* happens?

Suppose you move a total distance of 100 feet in a total time of 4 seconds. That means that the time it took to get to the half-way point, 50 feet, is half the time: 2 seconds. When you are at the one-quarter point, 25 feet, you are at the one quarter time: 1 second.

What you find is that every distance has a corresponding time associated with it. In fact, for any distance you mention (say 30 feet), I can give you the precise time you were at that point (30/4=7.5 seconds).

And we are guaranteed to go through those times! By whatever mechanism, we actually do go through the half-way time, the quarter-way time, etc. We do go through those times! And since each of those times is paired with a specific distance, we can *also* go through all those spatial points.

it is the very fact that both space and time have the same infinite size that makes motion possible.

We can do similar things with the other Zeno paradoxes. Take, for example, the paradox of Achilles and the Tortoise. They run a race where the Tortoise starts out 100 feet ahead. Achilles runs 10 times as fast, though. So we all know that Achilles will catch up and pass the Tortoise.

Zeno, however, argues as follows: when Achilles is at the place the Tortoise started, the Tortoise is 10 feet ahead. When Achilles runs that 10 extra feet, the Tortoise is 1 foot ahead. When Achilles runs that 1 foot, the Tortoise is still 1/10 of a foot ahead. Thereby Zeno claims that Achilles can never catch the Tortoise.

So, let's bring time into play and see how that affects things. Suppose for definiteness sake that Achilles runs 10 feet per second and the Tortoise 1 foot per second. When Achilles reaches the place where the Tortoise started, the Tortoise is, indeed 10 feet ahead. This has taken 10 seconds. When Achilles runs that 10 feet, the Tortoise is now 1 foot ahead and the clock has added a second to give 11 seconds since the start of the race. After another foot for Achilles, the Tortoise is 1/10 of a foot ahead and the clock reads 11.1 seconds.

What we realize is that the distance Achilles has gone is getting closer and closer to 111.1111... feet, the Tortoise has gone 11.111... feet, and the clock reads 11.1111.. seconds.

In other words, Achilles has gone 111 1/9 feet, the Tortoise has gone 11 1/9 feet and the clock has read 11 1/9 seconds. What we have found is the exactly instant that Achilles has passed the Tortoise! Far from showing Achilles cannot pass the Tortoise, this actually zooms in on the exact time and location where and when Achilles does so.

Again, the fact that time is just as divisible as space is what resolves the paradox and even gives an interesting way to find when the event of interest happens!

So, very far from being the cause of Zeno's paradoxes, infinite divisibility is exactly how those paradoxes are resolved! The fact that space and time are *equally* divisible is what allows motion to be possible. If one were finite and the other infinite, motion could not happen.

If both are finite, what we would find is that only very specific speeds would be allowed. But there is no reason to think we can't go half as fast as any speed we are currently going. So, if anything, this shows that an infinitely sub-dividable space and time is required for motion.

This, by the way, is a good reason to adopt at least infinite divisibility and thereby an actually infinite number of spatial points and time locations.