Can You Count to Infinity?
Part 9 of my Mapping the Kalam series.
Last time, we concluded our discussion of Craig & Sinclair’s first argument in favor of the beginning of the universe: the argument from the impossibility of an actual infinite, which looked like this:
2.11. An actual infinite cannot exist.
2.12. An infinite temporal regress of events is an actual infinite.
2.13. Therefore, an infinite temporal regress of events cannot exist.
Their second argument in favor of the beginning of the universe is their argument from the impossibility of the formation of an actual infinite by successive addition. It looks like this:
2.21. A collection formed by successive addition cannot be an actual infinite.
2.22. The temporal series of events is a collection formed by successive addition.
2.23. Therefore, the temporal series of events cannot be an actual infinite.
Notice that the second argument is independent of the first. Even if an actual infinite could exist, it could still be the case that an actual infinite could not be formed by successive addition. So if the temporal series of events is a collection formed by successful addition, and such a collection cannot be an actual infinite, then this argument provides independent proof in favor of the proposition that the universe began to exist in the finite past.
The authors begin with an important clarification:
…while it is true that 1 + 1 + … equals א 0., the operation of addition signified by “+” is not applied successively but simultaneously or, better, timelessly. One does not add the addenda in temporal succession: 1 + 1 = 2, then 2 +3 = 3, then 3 + 1 = 4, … , but rather all together. By contrast, we are concerned here with a temporal process of successive addition of one element after another.
One problem is that for any finite number n, we must admit that n + 1 is also a finite number. Thus א 0 has no immediate predecessor. And the problem with getting to an actual infinite by successive addition is not that we don’t have enough time. The problem is that with each successive addition, our total number is still n + 1, a finite number.
Thus, you cannot count to infinity. No matter how many numbers you count, you can always count one more before getting to infinity.
Can you count down from infinity?
Since you can’t count up to infinity, some have suggested you might count down from infinity. They say that although you can’t get to infinity by starting at a beginning point and counting forward, you could get to infinity “by never beginning but ending at a point, that is to say, ending at a point after having added one member after another from eternity.”
But before the present event could occur, the event immediately prior to it would have to occur; and before that event could occur, the event immediately prior to it would have to occur; and so on ad infinitum. One gets driven back and back into the infinite past, making it impossible for any event to occur. Thus, if the series of past events were beginningless, the present event could not have occurred, which is absurd.
Thus, we have good reason to accept
2.21. A collection formed by successive addition cannot be an actual infinite.
And premise 2.22. is a given if we accept the A-Theory of time, which most people do. So the only way to reject this argument in favor of the beginning of the universe is to reject the A-Theory of time. (But we shall talk about that later.)
This concludes our discussion of Craig & Sinclair’s two philosophical arguments in favor of premise 2 of the Kalam Cosmological Argument:
2. The universe began to exist.
Next, we consider Craig & Sinclair’s scientific arguments in favor of that premise.
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Craing & Sinclair: 2.21. A collection formed by successive addition cannot be an actual infinite.
lukeprog: And the problem with getting to an actual infinite by successive addition is not that we don’t have enough time.
But suppose the amount of time used was infinite. Reginald Selkirk
But before the present event could occur, the event immediately prior to it would have to occur; and before that event could occur, the event immediately prior to it would have to occur; and so on ad infinitum. One gets driven back and back into the infinite past, making it impossible for any event to occur. Thus, if the series of past events were beginningless, the present event could not have occurred, which is absurd.
I see the same failure to understand infinite time in all of these argument from Craig. He seems to be saying “the series of events couldn’t be beginningless, because it must have had a beginning.” His failure to understand does not prove a contradiction. Reginald Selkirk
The A-theory of time which the Kalam Cosmological argument ‘proves’ the necessity of a hard-stop beginning of the universe is seriously flawed. Nor do I get the feeling that Craig & Sinclair really understand what is implied in mathematics by ‘infinity’ – a behavior, a tendency, an ultimate goal, a description of where a function or series will end up, but not a number!
So Reginald Selkirk hit it on the head: failure to understand does not imply contradiction.
I wonder if Craig and Sinclair are familiar with the concept of a Manifold: a subset of a particular topology that for certain sets, the geometry appears Euclidian, while for the larger surface, there are clearly non-Euclidian features. The experimentation with non-euclidian geometries was directly related to Einstein’s formulation of special relativity. The application here is that the A-theory of time so favored by Craig & Sinclair is a manifold (in slightly curved space-time) with respect to the universe as a whole, in which time probably does not act as ‘consistently’ as we have evolved to expect it here. Rick
Suppose it was – the next question is: “when would you stop counting?” If you answer “never”, that affirms the point that is impossible. Thomas Reid
They understand that it is not a number. In fact in argument 2.1 they demonstrate the absurdities that would result if you treated it like one. The whole objection that actual infinites cannot exist is based upon the understanding that aleph-naught is not actually a number.
In 2.2 they are granting for the sake of argument that it possibly could be one, that’s all. They proceed from there to show that even if it could exist, it couldn’t be generated by counting.
Can you explain how this is relevant to whether or not actual infinites can be created by counting? Thomas Reid
I love the beginning of the BIG BANG timeline: “Time Begins.” It’s a little more complex than that, isn’t it? Shouldn’t it say: “Something pops out of nothing and time begins” ? I think that would be the ‘intellectually honest’ thing to say, wouldn’t it? We wouldn’t want to withhold the truth from the masses. We wouldn’t want to mislead them would we? Mark
“Something pops out of nothing and time begins”
Its a simgularity – we can’t see beyond it. There amy have been something – we cannot see.
” think that would be the ‘intellectually honest’ thing to say, wouldn’t it? We wouldn’t want to withhold the truth from the masses. We wouldn’t want to mislead them would we? ”
See above. Rich
Beginning to exist without a cause isn’t the same as coming from nothing. The Big Bang doesn’t imply that the universe came from a state of nonexistence. Derrida
Can someone briefly explain or link to an explanation of how infinity is used in mathematics? I don’t mean how it’s used in calculus, but how it’s used in trans-finite math (aleph-naught). It must be useful in mathematics, even if it cannot exist in the real world. (Even if it could, it certainly can’t be found as a scientific measurement that physicists might stick in an equation).
What can mathematicians do with a set with infinite members that they couldn’t do otherwise? Briang
Briang, it’s more like a classification scheme. For example, real numbers are of higher cardinality than integers. In a sense, they are “more infinite” in that a one-to-one mapping cannot be established between the two (i.e. an isomorphism). Time scale, I think, would share cardinality with the real number line, so would be 2^aleph naught cardinality. Time is then subject to theorems of the real number line. Different degrees of infinity (cardinality) are used to classify sets. For instance there is a whole class of theorems that apply to real numbers and not natural numbers, etc. Beelzebub
I’m beginning to suspect that there’s an ambiguity in this argument. Does “formed” mean “formed from nothing” or just something like “added to”? Obviously you can’t get from a zero-membered set to an infinity-membered set from nothing, but if that’s the argument, Craig is just begging the question. But if the universe is infinitely old, then it’s always been infinitely old, but that’s compatible with its history being added to though successive addition. Chris Hallquist
Well he means formed from nothing, but I don’t see how this is question-begging.
Start from the present day and count backwards each day in the past. You have an unlimited amount of time to do so, and you aren’t required to include the days it takes you to count (you don’t have to include tomorrow, for example). But if the past series of events is actually infinite, you will never stop counting. We could even have one or two of our friends helping out, or 90 trillion of our friends helping out, it can’t be done. But there’s nothing preventing your from trying, that is, starting at zero.
In this thought experiment, the “successive becoming” (moving from 999 to 1,000 for example) of each value in your incomplete sum is representative of the successive becoming of temporal events. The key to denying argument 2.2 would be to affirm that temporal becoming does not actually occur. Thomas Reid
It’s not related to counting infinities per se, but aimed at removing the foundation underlying the A-Theory of Time. If, in our local corner of the universe, we perceive time to be linear, while on a larger scale it’s a manifold, then there are other ways for time to have begun in the universe than a mere linear shot from any locus. This would undermine the premise of their argument that:
Time must have had a beginning because:
a) The A-Theory of time and
b) Everything has a starting point
If either of those premises is shown to be false, their conclusion may be bogus.
As an aside, we don’t know in any scientific sense that ‘Time Began’ at the big bang. What we do know is that beyond a certain distance we can’t detect EM waves, so we can’t say that anything happened before that time. Rick
But the argument doesn’t require time to be thought of us “linear”, or occuring in the same way throughout whatever exists. Linear is not equivalent to temporal becoming. It doesn’t have to be Euclidean, merely a progression along some path. So I’m still not clear on why a different shape to time undermines temporal becoming. Is this an idea of yours, or do you have some references where I could read more? Thomas Reid
I don’t have any references, and I’m arguing from a purely mathematical standpoint here. There are is a whole class of infinite series that have infinite terms but a finite sum. A simple example would be the sum of all fractions of the type 1/2^n. The whole infinite lot of them equals 2. But you’ll never get there by counting each sum made by incrementing n by +1. This doesn’t prove that it’s impossible to get to 2, just that you’ll have to count forever *in this particular manner* to get there. Now, suppose the universe had a beginning a finite amount of time past, and time progressed forwards starting with infinitesimal gaps, not unlike starting at n=infinity and progressing towards the present. Sure, it takes an infinite number of discrete steps, but we’re in the present – the universe got to now!
In fact, the whole idea of relativity (one of our better tools describing the universe at a large scale) is that space-time is a manifold whose topology depends on the amount of mass present. The key is that universal constant, the speed of light, by which we interrelate time and distance. Rick
Rick,
If I understand Craig’s argument, he realizes that you can divide time into infinitely smaller fractions. However, he argument is that you cannot take a fixed finite timespan and project it infinitely into the past. So you cannot have an infinite number of seconds (or days or years or months) in the history of the universe. Brian G
Be careful here. The limit of the series 1/2^n (where n=0,1,2…) is 2. You can keep adding that series up forever, and you will never get to 2. Furthermore, at every point in the series, if you treat the series as a set, you have a finite cardinality to your set. You will never have a completed set whose sum equals 2 and actually consists of an infinite number of items. Your set will never have cardinality aleph-naught.
Your example is applicable to any line of any length, or any discrete measure of time. You can chop it up into as many very small pieces as you desire, even at variable rates, but you will never have an infinite number of pieces. If the universe began a finite time ago, then no matter how small you would like to conceive of the time-step, there have been a finite number of discrete steps.
Although if you are agreeing that the universe began a finite time ago, then there’s no more need for discussion I suppose. Thomas Reid
Forgive me for intruding, but it seems to me that all these guys are really doing is repackaging Zeno’s Paradox. There’s nothing wrong with the logic per se, but we know from viewing the real world that the answer arrived at is neither meaningful nor useful. ThatOtherGuy
when are you going to address the scientific arguments? Bob
Why, next, of course! :) lukeprog
I must say that after reading all these arguments about infinity from Craig and Sinclair, I am falling into the camp that thinks they do not really understand the concept of infinity. I doubt anyone really has a grasp of how to reason about an actual infinity.
However, I do not think that inability to reason about actual infinities means it cannot exist. By analogy, I have heard it said (don’t remember where) that no one really has a grasp of the full implications quantum mechanics; the best people can do is understand the math. However, that does not make it so that an “actual quantum mechanics” cannot be realized. Erika