In this post, I want to address the claim made in the Kalam cosmological argument (and elsewhere) that "an actual infinite cannot exist". I'm not 100% sure what is meant by this, but it seems to be referencing the limit definition of infinities. So let's take a moment to discuss limits.

Infinity as a Limit

Let's say you have a series of numbers, for example:

hastily made with wolfram alpha |

^{1}/

_{2},

^{2}/

_{3},

^{3}/

_{4},

^{4}/

_{5},...

The pattern here is simply that the n

^{th}number is

^{n}/

_{n+1}. There are a few interesting things about this sequence, first, no matter how large 'n' is, the fraction is less than 1. Secondly, you can get as close to 1 as you wish by picking a large enough n. (for example, if you wish to be less than 0.01 away from 1, you can do it with n=100). We say that 1 is the limit of this sequence, the fact that we never get there is irrelevant, what we care about is that we can get as close as we wish.

So that's how a limit to a finite number works, what about limits to infinity? Simply put, we say a sequence has infinity as a limit if it grows without bound. In other words, for every number A there is some integer n such that every number in the sequence past the n

^{th}is bigger than A. Take for example the sequence of the powers of 2 (2,4,8,16,...), no matter how big a number you pick for A, there is some point when the rest of the sequence is bigger than A. So the limit of the sequence is infinity.

We 'Never' Reach Infinity

I'm pretty sure that this is what people are talking about when they say actual infinite doesn't exist. To borrow William Lane Craig's example in the post that started this whole thing, suppose I am piling up baseball cards one at a time and I continue doing this forever. We can describe the number of cards in the pile as a sequence (1,2,3,4,5,6,...), it's a pretty boring sequence, but it's trivial to see that the limit is infinity. However, there is no point in time when there are an infinite number of cards on the table, therefore, infinity doesn't exist. I am pretty sure this is the idea that is referenced with the statement "an actual infinity doesn't exist".

There is something to this, and it is a very interesting topic for investigation. We say the limit is infinity, and yet we never reach it at any point on the actual timeline. But this is the way it has to be, because once you pick a specific time on the timeline and examine it to see whether or not there is an infinite number of cards, you have cut off the sequence. You have changed the situation to a finite limit of finite numbers, of course the result will be finite. So it is true that in this situation there is never a moment where we have a pile containing an infinite number of things. In this setting perhaps it is reasonable to say that an actual infinity doesn't exist, but to extend that to all situations where an infinity might come up is a bit hasty.

Part of the difficulty here is that infinities can do some crazy things, and you have to be really careful with them, especially when you have multiple infinities interacting with each other, and that's sorta what we have here. "Never" and "forever" are both words that may or may not contain an implicit infinity, depending on how they are used. When we say that we will continue doing this "forever", do we really mean forever or do we mean for an arbitrarily large finite amount of time? What about never, is this referring to an arbitrarily large finite time or a limit? My intuition suggests that never is talking about an arbitrarily large finite time and forever is really talking about in the limit, in this sense we have a disconnect. [Although I'm a mathematician, so who knows how my intuition will line up with yours :)]

Let's Dispense with the Cards

Using the example of piling up cards does seem to illustrate some points, but it also creates some problems (Where do we get the cards? Does this assume infinity from the start? Where do we put them? Will they collapse into a black hole eventually? etc) So let's shift our example to just counting up time that has passed. We will count up seconds "forever", much of what we said above with the cards will transfer over, do we ever get to a point where we have counted an infinite number of seconds? No, we don't for the same reason as above, once we look at a point in time we have restricted ourselves to the finite. What if we step outside of our own universe and look at the whole thing all at once? I would say that in this case we can say that an infinite number of seconds is possible.

One of the things that humblesmith challenged me with is that my argument is circular, and I think this is a good spot to address it. In a sense I have started by assuming infinity is possible (when I said that we could count seconds forever), and now I have concluded that infinities can exist. In a sense he is right, this only works if forever exists which we assumed from the start. However, the key here is that I'm just trying to argue that this is a possibility. As far as I can tell, there are two possibilities, the universe ends at some point in the future (like a big crunch) or it doesn't (heat death for example). If the big crunch happens then my argument wouldn't work, and even from a perspective outside of our universe with a broad view we would not be able to observe an infinity. But if there is heat death (or some less boring eternity in our universe) then we could see infinity from the outside perspective. But again, I'm only arguing for this to be a possibility, so even if it doesn't work in our universe, I think we can argue that it could exist in

**some**universe. Arguing that something is possible is much easier than arguing that something is impossible.

Infinite Regress

Turtles all the way down (Photo credit: Wikipedia) |

Any Apologists Reading?

I have seen the claim that an infinite regress is impossible in many places. However, I have never actually seen it explicitly spelled out why that is the case. When it is hinted at, it always seems to be referencing the limit definition of infinity, so that is what I ran with. If there is a different argument as to why please let me know. I'd love to check it out, and if it gives me another excuse to write about infinities, all the better.

Great post. I love reading authors write about what's in their wheelhouse. What I never get is that apologists presuppose a past-infinite time-like dimension that their eternal God either existed within or was a part of God--believing THAT actual infinite how can they possibly argue against an idea like infinite regression. It's just special pleading.

ReplyDeleteThanks Grundy. That's a good point about the special pleading. I've had that argument a few times as well, they try to get around God having an infinite past by saying that he's timeless. My response would be that if God can be timeless (whatever that means) why couldn't the multiverse be timeless? I've never really gotten an answer to that.

DeleteI think it has something to do with things beginning, which is typically in the first premise to the argument.

DeleteBut isn't the argument that infinities are simply impossible? Whether God had a beginning or not is irrelevant to this point, it doesn't make sense to claim that God is infinite and that infinities are impossible.

Delete[sigh. not sure I want to repeat this, but here goes anyway.] If X has a beginning, it is caused, which requires something already actual to cause X to come to be. When X comes to be, it changes from potentially existing to actually existing. Things that change are not infinite, for infinites cannot increase into something they do not already have, or else they would not be infinite in the first place. Further, when X is caused to be, it cannot have something not already in the cause. Two infinites cannot be identical, for one would have to have something the other did not, or there would be no distinction. Therefore an infinite, in the same sense as the cause, cannot be created from things with beginnings. (this gets to the point that successive addition does not equal infinity, no matter how long it's added). Further, all movement from potential to actual involves things limited by potential. Anything with potential is not infinite, as evidenced by merely observing anything in creation, all of which has a limit due to it's limited potential. Therefore things with beginnings are not infinites. A thing without beginning does not have this limitation, for it is not caused, has no limiting potential, and is therefore purely actual, without any limiting potential. Matter is a limiting potential, so such a being has no matter. Such an infinite does indeed exist. This we call God. The argument about infinites is that actual things cannot be infinite if they have succession, matter, beginnings, or potential to change.

Delete@ humblesmith: How can this statement be true? "Two infinites cannot be identical" I mean if we are talking about infinite sand or infinite beans,yes I can agree. But if we are talking about the infinity concept then they are the same. So if something is infinite and you call it god,can I not just call it infinity.

DeleteThe problem is now you need to prove this infinity actually has importance outside of the concept of infinity.

humblesmith,

DeleteIt seems to me that you mean something very different from me when you use the word infinite. I think that is the source of our communication problem. Perhaps the difficulty here is that infinity can mean different things, and it is being used in different ways here. When we talk about infinite regress we mean one thing, when you say "God is infinite" you mean something completely different.

"Things that change are not infinite, for infinites cannot increase into something they do not already have, or else they would not be infinite in the first place."

This is a perfect example, you seem to be saying that infinities cannot grow, but in certain circumstances infinities can grow. Consider a line in a plane, it can be infinitely long but have no width. It is reasonable to say that the line is infinite, but it certainly has room to grow. I could thicken it up and we would have an infinitely long ribbon. It has changed, it has grown, and it was infinite before and it still infinite.

Now, I'm sure that when you are talking about infinite thing that can't change, you do not have this kind of infinite in mind, you are talking about something else entirely. That is what I'm trying to get at.

"Two infinites cannot be identical, for one would have to have something the other did not, or there would be no distinction."

If I replace infinity with omnipresent, I think I see what you are getting at here. If two different things were everywhere and everything, then they would actually be the same thing. Is this what you are getting at? However, if we mean infinity in a different sense, say the ribbon on a plane, there is plenty of room for multiple infinities.

"Further, all movement from potential to actual involves things limited by potential. Anything with potential is not infinite, as evidenced by merely observing anything in creation, all of which has a limit due to it's limited potential. Therefore things with beginnings are not infinites. A thing without beginning does not have this limitation, for it is not caused, has no limiting potential, and is therefore purely actual, without any limiting potential. Matter is a limiting potential, so such a being has no matter."

I'm going to be honest here, I have no idea what you are getting at. I think the root of the problem is your use of the word infinite.

"[sigh. not sure I want to repeat this, but here goes anyway.]"

I'm not sure if you think I'm just being thick, or stubborn, or stupid, or what. I'm really just trying to understand your argument, at the very least I think I have demonstrated the the wording makes it confusing and it seems reasonable to me that there is some miscommunication.

Sorry to jump in here. Was trying to follow along and I am missing something in your reasoning. Maybe you can correct me where I am going wrong, but this is how I was following your argument: Everything needs a cause and can’t come from nothing. Then you say things with beginnings cannot have infinities. You then jump to say that God has no beginning and is an infinity, but don’t really justify why this is or even how this might be different than a universe without a God. I think I am misunderstanding your argument, because the way I read it, it sounds like special pleading. Thanks!

DeleteI'm surprised people would claim that infinities can or cannot exist. I'm not sure we know enough about the universe to make a claim either way.

ReplyDeleteThere are trivial infinities that we construct ourselves using our invented math. Presumably there are an infinite number of directions to look into the night sky, for example. But, that's not really that interesting. I can also shoot a beam of light across an arc in that sky, and the line it traces out (if any two photons ever hit things really far away from us) will be traced out faster than the speed of light between them. But that's not very interesting either.

Some infinities that seem to interest are:

* locations in the cosmos ... is location in space-time discrete, or continuous?

* the eventual size of the accelerating, expanding cosmos (but is the acceleration itself decelerating?) and whether there's such a thing as time after its heat death

* the number of sub-particles... can quarks be divided? How far down?

* the number of universes... particularly if there's supposed to be one for every otherwise "random'" action of each particle

* whatever existed at the big bang, often described in terms of infinities

Infinity (and the math that results from using it) is a wonderful symbolic tool, but I don't know if we can ever say for sure whether it maps to something physical.

"I'm surprised people would claim that infinities can or cannot exist. I'm not sure we know enough about the universe to make a claim either way."

DeleteWell said, this is really the big point isn't it? There are certain situations where infinity can be argued for or against given various assumptions, but to completely say infinities are impossible (or alternatively that they absolutely have to exist) is pretty silly.

As to whether the universe is discrete or continuous, it is quite a ways outside my expertise, but the planck length is supposedly the theoretical minimum distance. I've heard it described as the resolution of the universe. If that is the case then at some point we can't subdivide things anymore and so we get finiteness.

http://en.wikipedia.org/wiki/Planck_length

As to whether the expansion of the universe is speeding up or slowing down, they were talking about this on titanium physicist a while ago. It was an episode about dark energy I think, they said it is expanding and the rate of expansion was accelerating, although if I recall correctly they also said that it was based on controversial ideas.

fun stuff :)

It is said that space is infinite, but we never reach the end of it. Is space not infinite?

ReplyDeleteSure, if we can never get to the end of something, we could say it is infinite for all intents and purposes.

DeleteI think the problem with infinity is that it is an abstract thought that we try and apply to a concrete problem like age of the universe. This is similar to quantum mechanics and particle/wave duality it is an abstract thought because as soon as you observe the duality the wave function collapses and you only observe one not a duality.

ReplyDeleteBut that is my two pence thoughts.

A potential infinity is possible, but an actual infinity is not. An example of potential infinity is time. Time began at the Big Bang and appears to be moving forward toward infinity (Heat Death). However, at any moment in time you can stop and you have a finite amount of time between t=0 and t=0+x. The potential future time is infinite, but it never becomes an actual infinite. Infinity is a concept or mathematical tool; it’s not a number.

ReplyDeleteOn reason an infinite regression is not possible is that you don’t get anywhere if you do not have a start. If there were an infinite number of past causes, you would not arrive at the present because there would continuously be one more cause before the cause that caused the universe. If there were really an infinite number of past causes, you wouldn’t even get to cause -10^10,000 before our universe’s cause. Otherwise, it’s not really infinity and it’s just a really big number.

Another reason is as humblesmith says; it’s circular reasoning. In order to claim that an infinite regression is possible you have to presuppose that an infinite regression is possible, so you are claiming that an infinite regression is possible if an infinite regression is possible.

“The infinite is nowhere to be found in reality. It neither exists in nature, nor provides a legitimate basis for rational thought. The role that remains for the infinite to play is solely that of an idea.” ~ David Hilbert

It's not even a potential infinity because in order to be infinite, it must go both ways in time without end. Lots of people will say that if you are born today and never die, your lifespan is effectively infinite. That's not true. Just in our universe alone, you were not around for 13.8 billion years. Since we know that this universe had a beginning, that means that anything in this universe could not have been around prior to 13.8 billion years ago, thus it cannot be infinite either. I agree with you, infinity is a mathematical concept, it has no actual application in reality.

DeleteFurther, the whole idea of infinite regression being a bad thing is based on our understanding of the physical laws of our particular universe, but since we know our universe had a beginning and came from somewhere, there's nothing to say that infinite regression has to be a part of the physical laws of whatever is outside of our universe. For all we know, infinite regression might be perfectly fine there and not violate any logical or physical laws. It's just an assumption, based on an incomplete understanding of the data set, that there's anything wrong with infinite regression at all.

Thanks for stopping by Rational Zealot. I'm glad you brought up the point that with an infinite past you can't get to the current moment. As far as I can tell, this argument assumes finiteness then concludes that the infinite is impossible. It is certainly possible that I am misunderstanding the argument somehow, if this is the case I hope you can set me straight.

Delete"you don’t get anywhere if you do not have a start. If there were an infinite number of past causes, you would not arrive at the present"

Why does there have to be a start? This seems to be an assertion rather than an argument to me. You seem to be asserting that there must be a start, which assumes finiteness because an infinite past doesn't have a beginning, then you conclude that infinite is impossible.

"If there were really an infinite number of past causes, you wouldn’t even get to cause -10^10,000 before our universe’s cause."

This, for example, seems to be assuming that there is a start, a first cause. Then by starting there we can't get to the current moment via a finite number of steps. But if there is an infinite regress there is no start. So this argument holds no water because the starting point that is critical to the argument doesn't exist.

If there is an infinite past, then every point on the timeline has an infinite number of points before it. However, between any two points there is finite distance between them. So when you say that we won't get to the current moment, I would ask you "from when?". From any point in the past, you can get to the current moment by some finite number of steps. The only point where we would have a problem is the point at negative infinity, but this point doesn't really exist. There is no first cause, that is sorta the whole point of an infinite regress.

If I'm misrepresenting your argument, please let me know. I really do want to understand this argument and why my response doesn't address the problem.

Cephus, you said

Delete"It's not even a potential infinity because in order to be infinite, it must go both ways in time without end."

You seem to be saying that an infinite thing would have to completely cover all moments in time. That's not typically what is meant by infinite. If there is an infinite future, why would taking 13.8 billion years away from it keep it from being infinite? It seems like you are talking about space-filling (or I suppose time-filling) as opposed to infinite.

@Hausdorff…..Infinity is not easy to discuss since it is a rather abstract concept and we are finite. The problem comes in when one starts using infinity as if it were a number, which is easy to do without realizing it. Hopefully I can phrase things better (aka longer) this time.

DeleteIf you start at zero and count in whole integers, you never reach true infinity because there is always one more number to count. The same is true if you flip the sequence to negative numbers. As you said, this puts in a starting point which is absent in infinity, but it you take away the starting point, you haven’t solved the problem. One cannot count from negative infinity to positive infinity, if that’s even a coherent statement. That’s the same as subtracting infinity from itself, which is undefined.

Let’s say the past is represented by negative numbers, zero is the present, and positive numbers are the future. Let’s say you never start counting but have been counting from an infinite past. An infinite amount of time later, you are still counting negative numbers. An infinite amount of time after that, you are still counting negative numbers. An infinite amount of time after that? Still negative numbers. To say otherwise means you haven’t really been counting from negative infinity, but have changed infinity into a number.

Then you have the weirdness of infinity when applied to reality. One example is Hilbert’s Hotel which is infinitely large and all of the rooms are full. If an infinite number of people come, it still has room to hold them. If an infinite number of aircraft carriers come with an infinite number of cars on them each containing an infinite number of people in them, there would still be enough room even though the hotel is full. If an infinite number of people leave, an undefined number of rooms remain full. If everyone in an even-numbered room leaves, of which there would be an infinite number, there would still be an infinite number of odd rooms occupied. If all but five people leave, which would also be an infinite number, there will only be five rooms filled. An infinite number of rooms were vacated in three different ways, yet in one scenario there are an undefined number of full rooms, in another there are an infinite number of full rooms, and in another there are only 5 full rooms.

The nature of infinity prevents it from being anything more than an idea, concept, or mathematical tool. What reason does one have to believe that infinity is possible in reality when it can neither be proven scientifically (our instruments can never measure infinity) nor is it logically coherent? It does not seem to be an adequate explanation for anyone using evidence to guide their beliefs.

"Infinity is not easy to discuss since it is a rather abstract concept and we are finite."

DeleteI agree with you 100% here. It's part of why I find it so fascinating and fun to talk about. It's also funny you mention hilbert's hotel, with all of this infinity talk, I've been toying with the idea of writing a post about hilbert hotel. :)

"Let’s say you never start counting but have been counting from an infinite past."

I guess this line is the key, when I read this, what I see is that at every point in time there is an infinite amount of counting that has already happened in the past.

"An infinite amount of time later, you are still counting negative numbers."

an infinite amount of time later

from when? It would appear that you are meaning that it is an infinite amount of time since the point at negative infinity, but we both agree that such a point doesn't exist."An infinite amount of time after that, you are still counting negative numbers."

This one definitely puzzles me. Even if we start at the theoretical point at negative infinity, if we count an infinite number of moments we will be somewhere on the timeline, if we then count an infinite number of points again we will now be at the theoretical point at positive infinity.

This appears to be an argument that I have seen before, which I can only make sense as starting at the point at negative infinity, and being unable to get onto the timeline. I'm not quite sure how to interpret this.

In relation to your last paragraph, I disagree with infinities being logically incoherent, but the rest I pretty much agree with. To assert that there

isand infinite past, or that I believe that such a thing definitely happened would be silly of me in my opinion. (it might not seem that way as I am arguing pretty hard for this, but I really do just want to argue that infinity is a possibility). For one thing, it requires there to be either a prior universe, or a multiverse. Obviously both of those are things we have no evidence for, and possibly can never have evidence for. But I also think ruling them out as logically impossible is a leap as well.“an infinite amount of time later from when?”

DeleteExactly! Infinity is a concept, not a number. There is no “from when”.

“It would appear that you are meaning that it is an infinite amount of time since the point at negative infinity, but we both agree that such a point doesn't exist.”

I do agree. There is no point or start with infinity.

“Even if we start at the theoretical point at negative infinity, if we count an infinite number of moments we will be somewhere on the timeline, if we then count an infinite number of points again we will now be at the theoretical point at positive infinity.”

Will we be at positive infinity? What’s infinity minus infinity?

“it requires there to be either a prior universe, or a multiverse”

If it’s possible, then I agree. I’m working on a few posts regarding the multi-verse, which if it exists, would be a literal Hilbert’s Hotel. Hopefully I’ll get them all posted before Christmas (I’m slow).

Thanks for the discussion!! It’s been enjoyable.

"Exactly! Infinity is a concept, not a number. There is no “from when”"

DeleteTrue enough, but you brought up the "infinite amount of time later" as the objection to an infinite past. I would argue that we can restrict ourselves to only being able to count from some finite point to some other finite point, and yet there could still be an infinite number of moments in the past (in the same way that there could be an infinite number of points in our future even though we can't count to infinity.)

I agree, it has been an enjoyable discussion. Can you do me a favor? When you get those posts written can you let me know so I don't miss them? Either post a new comment here or post to my twitter (@hausdorff80). Sounds like it would be right up my alley :)

And don't worry about being slow, I've got a new baby here, so it takes me forever to get anything done. One nice thing about text conversations is they can be asynchronous.

Will do! Congrats on the baby. I have two….7 & 9. It gets easier when the youngest turns 4. :)

DeleteTurns 4?!? You might as well say it will take a billion years :)

DeleteOh, I see. It won't work in this universe but it might work in the one next door. Well, if we ever get to the one next door, we can test it.

ReplyDeleteAs far as infinite regress goes, we know that this universe has a beginning so obviously there can't be one here. My understanding is that the point of this argument is to disallow an infinite regress even in a multiverse situation. Even if there was a universe that spawned ours, and another that spawned that one, and so on, there would have to be a first because an infinite regress is simply impossible in all situations. This is what I find to be an unfounded claim.

DeleteJust wanted to give a heads up to anyone following this thread, I made a new post about infinities today.

ReplyDeletehttp://hausdorffbb.blogspot.com/2013/12/even-in-infinite-regress-distance.html

I thought humblesmith and the rational zealot might be particularly interested as I quoted you guys.