Year: 2021

In Reasonable Faith, William Lane Craig seeks to show the absurdity of an actually infinity of things by appealing to surprises that await us at Hilbert’s Grand Hotel.

This is a thought experiment proposed by the German mathematician David Hilbert (1862-1943). In the thought experiment, Hilbert envisioned a Grand Hotel with an infinite number of rooms, all of which are full.

Hilbert then posed a series of scenarios that would allow the hotel to accept additional guests—both any finite number of guests, and even an infinite number of guests!

Craig poses four scenarios involving Hilbert’s Hotel, which I will summarize this way:

1. 1. A Single New Guest: The hotel is full and a new guest arrives. The manager has each current guest move to the next higher room, freeing up the first room for the new arrival.
   2. Infinite New Guests: The hotel is full and an infinite number of new guests arrive. The manager has each current guest move to the room whose number is double that of their current room. This puts all the existing guests in the even numbered rooms, leaving the odd numbered rooms free to accept the infinite number of new arrivals.
   3. Odd Numbers Check Out: The hotel is full and all the odd numbered guests decide to check out, leaving their rooms vacant. The manager then has the guests in the even numbered rooms move to the room whose number is half that of their current room. This fills up both the odd and the even numbered rooms, and the hotel is completely full again.
   4. Guests Above 3 Check Out: The hotel is full and all of the guests with room numbers above 3 check out, leaving only three guests in the hotel. Yet since the set {4, 5, 6, . . . } has the same “size” as the set {1, 3, 5 . . . }, it would seem that the same number of guests checked out in this example as in the previous one, but the resulting number of guests in the hotel was different.

Craig regards all of these as absurd and concludes both that such a hotel could not exist and that an actual infinity of things cannot exist.

How might we respond to this?

Yes, They’re Strange

My first response would be to say, “Yes, these are strange, surprising, and counter-intuitive. If you want, you can even call them absurd.”

But that’s not what I’m interested in. There are lots of true things that fit those descriptions.

What I’m concerned with is whether any of these situations entail logical contradictions that God could not actualize in some possible world, and it’s not obvious that they do.

What Infinity Means

It seems to me that part of the problem is that we have a persistent tendency to slip into thinking of infinity as if it is a particular, concrete, limited number. It’s not. By definition, it is unlimited. That’s what the term “infinite” means. But as long as we use labels like “infinity,” we tend to slip into thinking of this as if it were an ordinary, limited number rather than something of unlimited magnitude.

It can help to strip away the label and repose the question without it.

• • Suppose that I had an unlimited number of apples. Then you bring me a new apple, and I add it to my collection. How many apples do I have now? Well, it’s still an unlimited number. Adding a new item to an already unlimited collection isn’t going to change the status of the collection from unlimited to limited, so it’s still unlimited.
  • Suppose that you bring me a whole bunch of apples—an unlimited number of them—and I dump them into my collection. How many are there now? Again, adding even an unlimited number of apples to my already unlimited collection won’t change the status of my collection from unlimited to limited, so it’s still unlimited.
  • Suppose that you go through my apple collection and pull out all the odd numbered apples (let’s suppose that I’ve conveniently numbered them so I can always find the apple I want). How many are left in the collection? Pulling out every other apple from an unlimited number of apples would still leave an unlimited number there, so my collection is still of unlimited size. And I can re-number the apples I have left if I want.
  • Finally, suppose that you pull all of the apples out of my collection except the first three. How many do I have left? Three. “Why is the number different than when I pulled out the odd numbered ones?” you might ask. “Because of which ones you pulled out,” I reply. “The first time, you pulled out every other apple, but the second time you pulled out all of the apples but the first three. Of course, you’re going to get a different number left over at the end.”

What we’ve just done here is run through the same scenarios that Craig uses, only we’ve done it with apples instead of hotel rooms and we’ve done it using the intuitive term “unlimited” rather than less intuitive terms like “infinity” or “aleph-null,” both of which suggest a definite amount to our ears.

When we keep the fact that the amount is unlimited clearly in focus, the situation sounds quite a bit less counter-intuitive. The reason is that we aren’t losing sight of the not-limited amounts we are playing with.

But Could All This Really Exist?

Certainly not in our universe—not without God suspending the physical laws he has set up to govern it.

If a hotelier decided to build a hotel with an infinite number of rooms, he would be immediately confronted with the fact that there isn’t enough room on earth to house such a hotel, there aren’t enough construction workers to build it, and he doesn’t have an infinite amount of money.

Similarly, I can’t go to a store, or an orchard, or any number of stores and orchards, and buy an infinite number of apples. (Nor do I have an infinite amount of money—quite the opposite, in fact!)

So, both the hotelier and I would quickly discover that our projects are practically impossible (meaning: not possible given the practical restrictions we operate under).

Yet suppose the hotelier and I were to make go of our projects, somehow pulling in resources from off-world once earth’s supplies started to go low. At some point the hotel and the apple collection would collapse under their own mass.

If we kept loading material into them, they would become so massive that they would begin to fuse, and they would both turn into stars.

If we still kept loading material into them, they would become more massive yet and, eventually, millions of years later, go supernova and turn into black holes.

Such is the way of things, given the physical laws that operate in our universe.

Even if God were to intervene and miraculously allow the things to be built and not become black holes, we still wouldn’t be able to do the infinite amount of guest and apple shuffling involved in the thought experiments. We wouldn’t have the time!

So, I’m quite prepared to concede that, in addition to being practically impossible, the infinite hotel and the infinite apple collection are also physically impossible (meaning: not possible given the physical laws of our universe).

But that’s not our question. We’re not asking about practical possibility or even physical possibility. We’re asking about logical possibility.

How useful is a hotel metaphor?

Here I would like to introduce what I think is the main problem with Craig’s use of Hilbert’s Hotel to disprove the possibility of an infinite history: I just don’t think it’s relevant.

Even if you grant that Hilbert’s Hotel involves a logical impossibility, that doesn’t show that all actual infinities involve a logical impossibility.

We’ve already shown that there is an actually infinite set of mathematical truths, so that is logically possible.

And that actual infinity does not seem to be subject to the same kind of difficulties that Hilbert’s Hotel raises. Even if we were to start numbering the different mathematical truths the way the rooms in Hilbert’s Hotel are numbered, we couldn’t perform the kind of manipulations on them that the hotelier does.

We couldn’t have the mathematical truths move around the way the guests in the hotel do. We couldn’t make new mathematical truths appear or existing mathematical truths vanish.

We could change the numbers we apply to these truths, but that’s not the same thing. That’s just our labeling. It doesn’t affect the truths themselves.

The apple collection fares no better. We can’t move, add, or delete mathematical truths the way we can apples. Mathematical truths are just there.

And so it seems that the kind of puzzles we can generate with Hilbert’s Hotel simply do not arise with the actual infinity of mathematical truths.

This means, if we grant that Hilbert’s Hotel involves a logical contradiction, that we have to ask the question: How relevant is it to the idea of God creating an infinite history for the universe? Is the idea of an infinite history in the same category as Hilbert’s Hotel or in the same category as the set of mathematical truths?

Some Similarities

At first glance, you might think that an infinite history would go in the same category as Hilbert’s Hotel.

For one thing, Hilbert’s Hotel involves an infinite amount of space (the infinite rooms it contains), and an infinite history would involve an infinite amount of time. We could map the rooms of Hilbert’s Hotel onto the individual moments of time in the universe’s history.

Furthermore, the rooms in Hilbert’s Hotel have content (guests and the things they are doing in their rooms), and the moments of history have content (people and the things they are doing—or at least whatever God would have chosen to put in them).

But these similarities turn out to be rather superficial. In fact, you could map any countably infinite set onto any other countably infinite set. You could, thus, map the rooms of Hilbert’s Hotel onto something totally non-physical, like the mathematical truths that fit the form X + 1 = Y, like this:

• • Room 1 –> (1 + 1 = 2)
  • Room 2 –> (2 + 1 = 3)
  • Room 3 –> (3 + 1 = 4)
  • Room 4 –> (4 + 1 = 5)

And just as each room in the hotel has contents (guests), each slot in our sequence of mathematical truths has contents (the particular truth in question).

The real issue is whether the moments of history and their contents can be manipulated the way that the rooms and guests in a hotel can.

And they can’t be.

Time Can Be Rewritten?

Despite Doctor Who’s repeated assurances that time can be rewritten, this question has proved a real puzzler for philosophers and scientists.

Scientists are of different opinions about whether time travel to the past is physically possible. Einstein was startled when, in honor of his 70th birthday, Kurt Gödel gave him a proof showing that Einstein’s own equations would allow for the possibility of time travel if we are living in a certain kind of universe.[1]

So what would happen if you really were able to go back in time? Could you change history? Could you carry out the famous “grandfather paradox,” where you kill your own grandfather before the conception of your father, thus preventing your own birth? (And, by extension, preventing you from coming back in time to kill your grandfather.)

According to much of the thinking on the issue, there would be two possibilities:

1) You simply would not be able to kill your grandfather. According to this view, time has an as-yet undiscovered principle that prevents time paradoxes from happening. This was proposed in the 1980s by the Russian physicist Igor Novikov (b. 1935), and the proposed principle is known as the Novikov self-consistency principle.

2) You would be able to kill your grandfather, but in so doing you would create a new timeline. According to this view, you would be able to kill your grandfather, but this would not change your history. The timeline in which your father and you were born would still exist. That timeline has to still exist, because you need to leave at some point to come back in the past. What would happen if you kill your grandfather is you would cause a new timeline to branch off from the original one, and in the new timeline you would never be born. You would still be around, however, because your original timeline is still out there, undamaged.

If the first of these possibilities is correct then the Doctor is simply wrong. Time can’t be rewritten.

If the second of these possibilities is correct then the Doctor is right. –ish. Up to a point.

You could “change history” in the sense of causing a new timeline to branch, one in which you are never born. But you didn’t change your history. The timeline where you were born, grew up, and travelled back in time is still out there. It’s still real. It didn’t un-happen from an external perspective looking at the timelines.

How does this help us answer our question about an infinite history?

Forward, into the Past!

The ancients really don’t get enough credit, because since at least the time of the Greeks, people have been asking whether God could change the past.

The poet Agathon (c. 480-440 B.C.) wrote that:

For this alone is lacking even to God,
To make undone things that have once been done.

This statement is quoted and endorsed by Aristotle (Nichomachean Ethics 6:2).

The same view is taken up and endorsed by St. Augustine (c. 354-430), who wrote:

Accordingly, to say, if God is almighty, let Him make what has been done to be undone, is in fact to say, if God is almighty, let Him make a thing to be in the same sense both true and false [Reply to Faustus the Manichean 26:5].

And the same view is endorsed by St. Thomas Aquinas, who wrote:

[T]here does not fall under the scope of God’s omnipotence anything that implies a contradiction. Now that the past should not have been implies a contradiction. For as it implies a contradiction to say that Socrates is sitting, and is not sitting, so does it to say that he sat, and did not sit. But to say that he did sit is to say that it happened in the past. To say that he did not sit, is to say that it did not happen. Whence, that the past should not have been, does not come under the scope of divine power [Summa Theologiae I:25:4].

Notice that both Augustine and Aquinas identify the same reason for God not being able to change the past: It involves a logical contradiction.

Does it?

From Here to Eternity

The view that I’ve advocated in this paper—that God is outside of time and that, consequently, the B-Theory of time is true—indicates that God cannot change the past—at least not in the way relevant to our discussion.

To illustrate this, let’s think about a particular moment in my own past, which I have already referred to: the moment when I was five years old and, in my grandmother’s kitchen, reached for a broom, causing a Coke bottle to explode and injure my knee.

That moment is present to God in eternity.

In the eternal now, he is creating that moment in history, and, by my free will, I am grabbing the broom and causing the Coke bottle to explode.

Could God change history so that this doesn’t happen? Could he, for example, give my grandmother a sudden inspiration a few moments earlier so that she can intervene and prevent me from grabbing the broom?

No.

Why not?

Because that moment is present to him in the eternal now. It’s real. He can’t undo it, because there is no time where he is. He can’t first let it happen and then undo it. That would imply the passage of time for God. It would imply more than one moment in the eternal now, in which case God would not be in eternity but in time.

What God could do (I assume) is create a second timeline,[2] in which my grandmother does stop me from breaking the Coke bottle, but that would leave intact the original moment in which I did break it. That moment is still there, still present to God in eternity, and thus still real.

If Akin is right about this then so were Agathon, Aristotle, Augustine, and Aquinas: God cannot undo history.

Once an event has happened in time, it’s set in eternity.

This has important implications for the question of whether he could make an infinite history.

Not Like Hilbert’s Hotel

If God can’t change history then time is not like Hilbert’s Hotel.

God cannot, in the eternal now, cut up, rearrange, and reshuffle the moments of history the way Hilbert’s hotelier moved guests around. If Moment 2 follows Moment 1 in time, then that is the way they are arranged before God in eternity. He can’t swap their order because in eternity there is no time for him to swap them in.

God cannot, in the eternal now, create or delete additional moments in time beyond those he has already created, the way I could add or subtract apples from my collection.

There is no time in which God could do these things, because that is the way eternity works by definition. It is thus logically impossible.

That means that the kind of puzzles that arise with Hilbert’s Hotel simply do not arise with time from God’s perspective.

An actually infinite history for the universe thus would go in the same category as an actually infinite number of mathematical truths. It does not give rise to Hilbert paradoxes.

Neither does an actually infinite future for the universe, which is why God is able to create one of those—as he has.

Therefore, even if one grants that Hilbert’s Hotel involves a logical contradiction and could not be realized by an omnipotent God, it does not matter. Viewed from the eternal perspective, an infinite amount of time (past or future) does not generate the problems of Hilbert’s Hotel and so is not in the same category.

[1] This solution is known as the “Gödel metric.” For a good popular introduction to the subject of time travel from the perspective of contemporary physics, see The Physics of the Impossible by Michio Kaku, who is a professor of theoretical physics at City University of New York.

[2] Because God is in eternity, this second timeline, and any others he might create, would all exist simultaneously for him in the eternal now, just as our timeline does. From the eternal now, he would be simultaneously creating all of the histories that exist.

We know that God created the universe a finite amount of time ago, but defenders of the Kalaam cosmological argument say that God had to do it this way. He had no other choice.

William Lane Craig has proposed the following argument to support this claim:

1) An actually infinite number of things cannot exist.

2) A beginningless series of events in time entails an actually infinite number of things.

3) Therefore, a beginningless series of events in time cannot exist.

Depending on your view of time, there are potential problems with the second premise—as I’ve written elsewhere.

However, here I’d like to consider the first premise.

Is it true that an actually infinite number of things can’t exist?

What’s an “Actual” Infinite?

We need to be aware of the difference between what philosophers and mathematicians call “potential” infinities and “actual” infinities.

• • Something is potentially infinite if it goes on endlessly, but there is no frame of reference in which all of its elements exist.
  • Something is actually infinite if it goes on endlessly and there is a frame of reference in which all its elements exist.

For example, suppose that you have a machine that makes cubes. Today it makes a cube, tomorrow it makes a cube, and it keeps on like that endlessly. No matter how many days you go into the future, there will still be a finite (limited) number of cubes.

From within the perspective of time, there is no day where the “infinity-eth” cube pops out of the machine, because “infinity” is not a number on the number line. There is no “number just before infinity,” and so you can’t count to infinity.

Yet the series of cubes that the machine will make is endless, which is what “infinite” means—unlimited or unending (Latin, in- “not” + finis = “limit, end”).

So, while the number of cubes you have grows toward infinity, it never gets there. That’s why the cubes would be said to be potentially infinite rather than actually infinite—because they don’t all exist at the same time.

But suppose you didn’t have a cube-making machine. Suppose instead that God decided to create an infinite number of cubes all at once. Bam! It’s done. All in a flash.

In this case, you would have an infinite number of cubes—that all exist at once—and so that would be an actually infinite set of cubes.

The key point to remember is that both potential infinites and actual infinites involve limitless numbers of things. The difference is that in a potential infinity these elements don’t all exist at once, while in an actual infinity, they do.

Can Actual Infinities Exist?

While many authors assert that actual infinities can’t exist, we should test this. Can we think of anything actually infinite?

How about numbers?

We often represent the set of natural numbers like this: {0, 1, 2, 3 . . . }. The reason we put the ellipsis (the three dots) at the end is to say that this series of numbers goes on forever in the same way it began, with one number after another, with no end to them.

That’s an infinite set!

And all those numbers already exist. It’s not like there’s a mathematician somewhere inventing new numbers in his workshop.

We may make up names for new numbers—like the number googol, which is 10-to-the-power-of-100, or googolplex, which is 10-to-the-power-of-googol—but we didn’t create these numbers. We only named them.

Even before they were thought of or named, it would remain true that googol multiplied by 2 is 2 googol, that googol minus google is 0, and that googol divided by google is 1.

So, it appears that an infinite quantity of natural numbers exists, whether we’ve thought of or named them or not.

Since the set of numbers does not grow with time—only our knowledge of them does—the set of natural numbers is an actually infinite set, as mathematicians commonly acknowledge.

The Truth of the Matter

Numbers are not the only actual infinity we can think of. There also are truths (facts), as we can easily see:

• • It is true that 1 + 1 = 2.
  • It is true that 2 + 1 = 3.
  • It is true that 3 + 1 = 4.
  • And so on.

So, not only does an actually infinite quantity of numbers exist, an actually infinite quantity of truths also does.

How might a defender of the Kalaam argument respond to this?

The Nature of Numbers and Truths

It’s easy to point out that there is a difference between numbers and truths and the kind of objects we see in the world around us.

For example, if I have two cubes that I’m holding in my hands, I’m physically touching them. But I don’t seem to be touching the number 2. Numbers aren’t physical objects you can see, hear, or touch.

Neither are truths. It may be a truth that Abraham Lincoln died in 1865, but I can’t hold this truth in my hands like an apple or put a ruler beside it and measure how long it is.

Things like numbers and truths seem to be in a different category than things like cubes and apples. Things in the first category are often called abstract objects, while those in the latter are often called concrete objects. We also might call them physical objects.

The sciences have given us a lot of information about how physical objects work, but we can’t use science to investigate abstract ones. They lie in the realm of philosophy, and among philosophers there are different views about their nature.

Some philosophers (known as anti-realists) deny that abstract objects are real and propose other ways of understanding them. Other philosophers (known as realists) hold that they do exist, but again there are different understandings (e.g., do abstract objects exist in an abstract realm of some kind? do they only exist in physical objects? are they based in the mind of God?).

Options for the Kalaam Defender

A Kalaam defender could adopt an anti-realist position and say that things like numbers and truths simply do not exist, which would mean that there aren’t actual infinities of these—because numbers and truths don’t exist in the first place!

But this seems hard for many to imagine, including various supporters of the Kalaam argument.

There is another option, which is to draw a line between the two classes of objects and say something like, “Look, whether or not actual infinities of abstract objects exist (maybe they do; maybe they don’t), that’s not what I’m talking about. When I say that actual infinities can’t exist, I mean that they can’t exist concretely, in the physical world.”

A person taking this position could acknowledge that actual infinities of abstract objects can exist. What he disputes is that actual infinities of concrete, physical ones can.

Why Not?

The question would be: Why not? Why can’t actual infinities of physical objects be real?

This question is particularly acute from a Christian perspective, since the Christian faith holds that God exists and that he is omnipotent, which means that he can create anything that does not involve a logical contradiction.

So, let’s do a thought experiment:

Imagine a cubic foot of empty space that has a single hydrogen atom in it. If we can imagine that, so can God.

Now imagine a second cubic foot of empty space sitting right next to it, also with a hydrogen atom in it. God can imagine that, too.

Now put a third cubic foot of space next to that, also with a hydrogen atom, so that we have a row of three.

Then imagine a fourth, a fifth, a sixth, and so on. God can imagine each of these as the line of cubic feet extends off into the distance.

In fact—due to his omniscience—God can imagine any number of such units. Unless there is a logical contradiction involved, God could imagine a cubic foot of space with a hydrogen atom for every natural number.

And so, God can imagine an actually infinite volume of space, with a single hydrogen atom in each cubic foot.

Either this scenario involves a logical contradiction or it doesn’t. If it doesn’t, then God can imagine it, since God can do anything logically possible.

And it does not appear to involve a contradiction. As even many defenders of the Kalaam argument admit, the mathematics of infinity are consistent and do not contain logical contradictions.

This situation isn’t like saying, “Suppose God imagines a four-sided triangle.” Not even God can visualize that, because “four-sided triangle” is a contradiction in terms.

“Four-sided” and “triangular” mean different and contradictory things. This expression is just word salad—not something that is actually meaningful.

But our volume of space isn’t like that. Even our puny minds can imagine a row of cubic feet with hydrogen atoms in them. God’s mind is infinite, and so he can imagine the line of cubic feet extending on endlessly—and there doesn’t seem to be a logical contradiction involved in him doing that.

If there’s not, then we’re ready to add a new element to our thought experiment.

“Let There Be Space!”

Because of his omnipotence, God can do anything that doesn’t involve a logical contradiction. So, if God can imagine something, he can also create it.

As a result, if the idea of an infinite row of cubic feet of space—each with a hydrogen atom—doesn’t involve a logical contradiction, then God can make it real.

God thus could create an infinite number of hydrogen atoms and an infinite volume of space to contain them.

If God chooses, actual infinities of physical things could exist!

The only way to avoid this would be to say that, even though the idea of finite space with finite atoms is logically coherent, a contradiction in terms is generated if we extend this to infinity.

In that case, God couldn’t imagine or create this any more than he could a four-sided triangle.

But in that case, I want to know: What’s the contradiction?

As we’ve discussed, a Christian who understands God’s omnipotence should affirm that God can make something unless it is shown to involve a logical contradiction.

The Kalaam defender thus needs to name the contradiction: Which are the specific terms that contradict?

And it won’t do to change the scenario and pose some other one where a new logically contradictory entity is subtly introduced. That happens all too often.

Changing the scenario is what you do when you can’t deal with the current one.

Neither is it sufficient to say, “Hey, infinities have weird properties.” Yes, they do. That doesn’t mean they’re beyond the reach of God’s omnipotence.

So please, deal with the thought experiment I’ve described, flesh out its terms in detail, and name which terms contradict each other—just like how I pointed out that “four-sided” and “triangular” contradict.

Either that or be prepared to acknowledge that God has the power to create actual infinities of physical objects.

We know from Scripture that God created the world a finite amount of time ago, but was that the only option he had? Could God have created a world with an infinite history?

Defenders of the Kalaam cosmological argument claim that he couldn’t have done so.

To show that, they would need to demonstrate that the idea of an infinite history involves a logical contradiction.

Some recent attempts to do this involve paradoxes that have been proposed by different authors.

Let’s look at a couple and see what we can learn.

The (Squished) Grim Reaper Paradox

The core of what has become known as the Grim Reaper paradox was proposed by Jose Bernardete, and the argument has taken different forms.

Here I’ll present a version that is close to the original. I’ll refer to it as the “squished” version for reasons that will become obvious.

Suppose that a guy named Fred is alive at 12:00 noon.

However, there are an infinite number of grim reapers waiting to kill him. For the sake of convenience, we will give the reapers names based on the negative numbers, with the last reaper being Reaper 0.

If Fred is still alive at 1 p.m., Reaper 0 will encounter him and kill him. However, before that, Reaper -1 will encounter Fred at 12:30 p.m. and kill him. Even before that, Reaper -2 will encounter Fred at 12:15 p.m. and kill him. But Reaper -3 will encounter him at 12:07:30 p.m., and so on, with each reaper set to encounter (and kill) Fred in half the remaining distance back to noon.

Which reaper will kill Fred?

The way the situation has been set up, we have a paradox. Reaper 0 should not kill Fred, because Fred should already have been killed by Reaper -1. But this reaper shouldn’t kill him either, because he should have been killed by Reaper -2. And Reaper -3 should have killed him before that, and so on.

It thus looks like Fred can’t possibly survive past 12:00 noon, but it’s impossible to name which reaper kills him. Paradox.

I called the above version “squished” because it squishes the infinite series of grim reapers into a single hour. But we don’t have to do it that way. We’ll see another version later.

The Problem with the Paradox

The resolution of this paradox is fairly straightforward. It has envisioned a situation where Fred begins alive and then will be killed by the first grim reaper he encounters.

The problem is that—if the series of grim reapers is infinite—then it must have no beginning.

To suppose that an infinite series of whole numbers has both a first and last member involves what I’ve called the First-and-Last Fallacy.

• • Infinite series can have no beginning ( . . . -3, -2, -1, 0)
  • They can have no end (0, 1, 2, 3 . . .)
  • Or they can lack both a beginning and an end ( . . . -3, -2, -1, 0, 1, 2, 3 . . . )

But if a series has both a beginning and an end, then it’s finite.

The series of reapers set to kill Fred has an end—Reaper 0—but if that’s the case, it cannot have a beginning.

This means that there is no first grim reaper that Fred encounters, just as there is no “first negative number.”

The idea of a first negative number involves a logical contradiction, and therefore the (Squished) Grim Reaper paradox is proposing a situation that cannot exist.

The (Spread-Out) Grim Reaper Paradox

To make the Grim Reaper Paradox more relevant to the Kalaam argument, some have proposed a new version that spreads out the grim reapers over an infinite history rather than squishing them into a single hour.

We can put the spread-out version like this:

Suppose that Fred has always been alive, all the way through an infinite past.

Suppose that there is an infinite series of grim reapers set to kill Fred, and they are set to kill him in sequence on New Year’s Day.

On New Year’s Day this year, Reaper 0 is set to kill him if he is still alive. But on New Year’s Day last year, Reaper -1 was set to kill him, and Reaper -2 the year before that, and so on.

Which reaper kills Fred?

Exactly the same paradox results. Each reaper should not be able to kill Fred because a previous reaper should already have done the dirty work.

But the problem is the same: The situation is set up so that the only reaper that could kill Fred is the first one in a beginningless series, and a beginningless series has no first element.

The spread-out version of the paradox thus has the same flaw that the original did: It proposes an entity that involves a logical contradiction and so can’t exist.

Application to the Kalaam Argument

Kalaam defenders use paradoxes like this in an attempt to undermine the idea of an infinite history.

For example, on the rhetorical level, the strategy can work like this:

• • They encourage their audience to imagine a set of circumstances that could exist and that don’t involve a logical contradiction (e.g., Fred exists, a grim reaper is scheduled to kill him at some time).
  • They multiply the circumstance extending finitely into the past (suppose there are some other grim reapers who were also set to kill him previously).
  • They extend this infinitely into the past, generating a contradiction.
  • They point out this contradiction.
  • Finally, they assert that, since the infinite extension into the past caused the contradiction, we must reject the idea of an infinite past.

This reasoning is mistaken, because it isn’t the infinite history that’s the problem. It’s the fact that you’ve proposed a first element in a beginningless series.

Consider the following scenario:

• • Suppose that some guy named Fred exists.
  • Suppose that he existed last year.
  • And the year before that.
  • And that he’s always existed, with no beginning.

That involves an infinite history, but it doesn’t have a contradiction in it—because it does not propose a first element in a beginningless series.

It’s only when you introduce the latter that a contradiction occurs. So, it isn’t the infinite history that’s the problem but the impossible first element. Grim Reaper-like paradoxes thus are disguised forms of the First-and-Last Fallacy.

World Without End

Another way of seeing why the strategy used by Kalaam defenders is problematic is by flipping the arrow of time and considering a mirror image of the paradox that deals with the future:

Suppose that Fred is alive today and will remain alive as long as a grim reaper doesn’t kill him.

Suppose that there is an infinite series of grim reapers to kill him in the future and who we will name using the positive numbers, beginning with 0. Reaper 0 encounters Fred today, Reaper 1 encounters him tomorrow, Reaper 2 the day after, and so on.

But the reapers have agreed that the honor of killing Fred will go to the reaper with the highest number.

Which reaper kills Fred?

The way this scenario has been set up, the only reaper that can kill Fred is the last reaper.

But there can be no last reaper in a series that has no end, just as there can be no first reaper in a series that has no beginning.

The Future Reaper Paradox proposes the same kind of logically contradictory entity that the original paradoxes did.

And the problem is not the fact that the future is infinite in the scenario. The infinite future itself does not involve a contradiction, because it does not propose there being a last element in the series of days stretching into the future.

Indeed! From a Christian point of view, an infinite future is exactly what awaits us. The Christian faith teaches that God will give us endless life and there will be no day on which we pass out of existence. As a result, an orthodox Christian is committed to the idea of an infinite future.

It’s not the endlessness of the future that’s a problem in the above scenario, but the idea of that future containing a logically contradictory entity like the last member of an endless series of reapers.

In the same way, the idea of an infinite past is not a logical contradiction but the idea of a first reaper in a beginningless series.

The Underlying Assumptions

The paradoxes recently proposed in support of the Kalaam argument (and there are many) share a common set of assumptions:

1. 1. Suppose some past scenario (P) that involves a first member.
   2. Suppose some past scenario (P’) that does not have a first member.
   3. Suppose that P and P’ are the same scenario.

By contrast, corresponding future-oriented paradoxes have these assumptions:

1. 1. Suppose some future scenario (F) that involves a last member.
   2. Suppose some future scenario (F’) that does not have a last member.
   3. Suppose that F and F’ are the same scenario.

The way the paradoxes are fleshed out and expressed varies, and this can disguise the fact, but they all share the same set of assumptions.

As a result, they set up logical contradictions, but this does not tell us which of the premises must be rejected to diffuse the contradiction.

This is true of any such paradox. Consider the following geometrical one:

1. 1. Suppose there is some closed geometrical shape (S) that has three sides.
   2. Suppose that there is some closed geometrical shape (S’) that has four sides.
   3. Suppose that S and S’ are the same shape.

This scenario sets up the idea of a four-sided triangle, which is logically impossible. But in resolving the paradox, we don’t have to reject any particular premise.

We could reject the idea that S (the three-sided shape) exists; we could reject the idea that (S’) the four-sided shape exists; or we could reject the idea that S and S’ are the same and then conclude that there must be two shapes. Each of these is a legitimate option.

As a result, when considering past- or future-oriented paradoxes, we don’t have to reject the idea of an infinite past or an infinite future, because neither of these concepts generates a logical contradiction on its own.

It is only when we introduce a logically contradictory entity into these scenarios—like the first member of a beginningless series or the last member in an endless series—that a paradox results.

Those are the entities that need to be rejected, and the recently proposed paradoxes do not disprove either the idea of an infinite past or an infinite future.

In recent years, one of the most popular arguments for the existence of God has been the Kalaam cosmological argument.

Ultimately, I think this argument is successful, but many of the ways it has been employed are unsuccessful.

It is an argument that needs to be used carefully—with the proper qualifiers.

Stating the Argument

We can state the Kalaam argument like this:

1) Everything that has a beginning has a cause.

2) The universe has a beginning.

3) Therefore, the universe has a cause (which would be God).

Is this argument valid? Is it sound?

Valid arguments are ones that use a correct logical form—regardless of whether their premises are true. The Kalaam argument falls into this category, which is not disputed by its critics.

If a valid argument has true premises, then its conclusion also will be true. Valid arguments that have true premises are called sound arguments, and I agree that the argument’s premises are true:

1) It is true that whatever has a beginning has a cause.

2) And it is true that the universe has a beginning (approximately 13.8 billion years ago, according to Big Bang cosmology).

Since the Kalaam argument is valid and has true premises, it is a sound argument.

Using the Argument Apologetically

The Kalaam argument is sound from the perspective of logic, but how useful is it from the perspective of apologetics? There are many arguments that are sound, but sometimes they are not very useful in practice.

For example, in their famous book Principia Mathematica, Bertrand Russell and Alfred North Whitehead spend the first 360 pages of the book covering basic principles that build up to them rigorously proving that 1 + 1 = 2.

While their book is of interest to mathematicians, and their proof extremely well thought-out, it is so complex that it is not of practical use for a popular audience. For ordinary people, there are much simpler ways to prove that 1 + 1 = 2. (If needed, just put one apple on a table, put another one next to it, and count the apples both individually and together.)

Complexity is not the only thing that can limit an argument’s usefulness. Another is the willingness of people to grant the truth of its premises. Here is where some of the limitations of the Kalaam argument appear. While it is very simple to state and understand, defending the premises is more involved.

The First Premise

The first premise—that everything that has a beginning has a cause—is intuitive and is accepted by most people.

Some object to this premise on philosophical grounds or on scientific ones, such as by pointing to the randomness of quantum physics.

Both the philosophical and the scientific arguments can get technical quickly, but a skilled apologist—at least one who is actually familiar with quantum mechanics (!)—would still be able to navigate such objections without getting too far over the heads of a popular audience.

This—plus the fact that a popular audience’s sympathies will be with the first premise—mean that the argument retains its usefulness with a general audience.

The Second Premise

The second premise—that the universe had a beginning—is also widely accepted today, due in large part to Big Bang cosmology. A popular audience will thus be generally sympathetic to the second premise.

That’s apologetically useful, but we need to look more closely at how the second premise can be supported when challenged.

Since “The Bible says the universe has a beginning” will not be convincing to those who are not already believers, there are two approaches to doing this—the scientific and the philosophical.

The Scientific Approach

For an apologist, the approach here is straight forward: For a popular level audience, simply present a popular-level account of the evidence that has led cosmologists to conclude that the Big Bang occurred.

On this front, the principal danger for the apologist is overselling the evidence in one of several ways.

First, many apologists do not keep up with developments in cosmology, and they may be relying on an outdated account of the Big Bang.

For example, about 40 years ago, it was common to hear cosmologists speak of the Big Bang as an event that involved a singularity—where all matter was compressed into a point of infinite density and when space and time suddenly sprang into existence.

That view is no longer standard in cosmology, and today no apologist should be speaking as if this is what the science shows. Apologists need to be familiar with the current state of cosmological thought (as well as common misunderstandings of the Big Bang) and avoid misrepresenting current cosmological views.

Thus, they should not say that the Big Bang is proof that the universe had an absolute beginning. While the Big Bang is consistent with an absolute beginning, cosmologists have not been able to rule out options like there being a prior universe.

One way apologists have dealt with this concern is to point to the Borde-Guth-Vilenkin (BGV) theorem, which seeks to show that—on certain assumptions—even if there were one or more prior universes, there can’t be an unlimited number of them.

It’s fair to point to this theorem, but it would be a mistake for an apologist to present it as final proof, because the theorem depends on certain assumptions (e.g., that the universe has—on average—been expanding throughout its history) that cannot be taken for granted.

Further, apologists should be aware that authors of the theorem—Alan Guth and Alexander Vilenkin—do not agree that it shows the universe had to have a beginning. Guth apparently believes that the universe does not have a beginning, and Vilenkin states that all the theorem shows is that the expansion of the universe had to have a beginning, not the universe itself.

It thus would misrepresent the BGV theorem as showing that the scientific community has concluded that the universe had to have a beginning, even if it were before the Big Bang. (It also would be apologetically dangerous and foolish to do so, as the facts I’ve just mentioned could be thrown in the apologist’s face, discrediting him before his audience.)

Most fundamentally, the findings of science are always provisional, and the history of science contains innumerable cases where scientific opinion as reversed as new evidence has been found.

Consequently, apologists should never sell Big Bang cosmology—or any other aspect of science—as final “proof.”

This does not mean that apologists can’t appeal to scientific evidence. When the findings of science point in the direction aspects of the Faith, it is entirely fair to point that out. They just must not be oversold.

The Philosophical Approach

Prior to the mid-20^th century, Big Bang cosmology had not been developed, and the scientific approach to defending the Kalaam argument’s second premise was not available.

Consequently, earlier discussions relied on philosophical arguments to try to show that the universe must have a beginning.

Such arguments remain a major part of the discussion today, and new philosophical ways of defending the second premise have been proposed.

Authors have different opinions about how well these work, but in studying them, I find myself agreeing with St. Thomas Aquinas that they do not. Thus far, I have not discovered any philosophical argument—ancient or modern—that I thought proved its case.

This is not to say that they don’t have superficial appeal. They do; otherwise, people wouldn’t propose them.

But when one thinks them through carefully, they all contain hidden flaws that keep them from succeeding—some of which are being discussed in this series.

I thus do not rely on philosophical arguments in my own presentation of the Kalaam argument.

Conclusion

The Kalaam cosmological argument is a valid and sound argument. It does prove that the universe has a cause, which can meaningfully be called God.

As a result, it can be used by apologists, and its simplicity makes it particularly attractive.

I use it myself, such as in my short, popular-level book The Words of Eternal Life.

However, the argument needs to be presented carefully. The scientific evidence we currently have is consistent with and suggestive of the world having a beginning in the finite past, though this evidence must not be oversold.

The philosophical arguments for the universe having a beginning are much more problematic. I do not believe that the ones developed to date work, and so I do not use them.

I thus advise other apologists to think carefully before doing so and to rigorously test these arguments: Seek out counterarguments, carefully consider them, and see if you can show why the arguments don’t work.

It is not enough that we find an argument convenient or initially plausible. We owe it to the truth, and honesty in doing apologetics compels us not to use arguments just because we want them to be true.

“In the beginning, God created the heavens and the earth.” While the world definitely had a beginning, there’s a question of whether we can prove this by reason alone (i.e., by philosophical arguments).

Defenders of the Kalaam cosmological argument often use an argument like this one, which is found in William Lane Craig’s book Reasonable Faith:

1) An actually infinite number of things cannot exist.

2) A beginningless series of events in time entails an actually infinite number of things.

3) Therefore, a beginningless series of events in time cannot exist.

I have a problem with the first premise, but that’s a topic for another time. Here I’d like to look at Craig’s second premise.

Is it true that a beginningless series of events entails an actually infinite number of things?

At first glance, the answer would seem to be yes, but the reality is more complex.

The Nature of Time

The answer depends on your view of time. Here we need to consider two major theories of time, which are known as eternalism and presentism.

Eternalism holds that all of time exists. The past, the present, and the future are all real from the ultimate perspective—that is, from the eternal perspective outside of time. We may only experience history one bit at a time, but from the “eternal now” that God dwells in, all moments of time are equally real.

Presentism (as we will be using the term) holds that, from the ultimate perspective, the only time that exists is right now—the present. The past used to be real, but it is no longer. And the future will exist, but it does not yet. Since neither the past nor the future are real, they do not exist in any sense of the word. If you asked God—from his ultimate perspective—“What is real in the created order?” he would answer, “Only the present.”

The Eternalist Option

Supposing that eternalism is true, Craig’s second premise would be true. From the eternal perspective outside of time, God would see an infinite series of past events laid out before him.

Or, if you wish to avoid the question of how God’s knowledge works then, as the Creator, God would be causing that infinite series of past events to exist.

They would all be equally real—equally actual—from his perspective, and—since they have no beginning—they would be infinite. Being both actual and infinite, the events of a beginningless history would represent an actual infinity. Thus, the second premise would be true.

But for a classical Christian theist, there would be a problem, because Christianity teaches that God will give people endless life. While human beings may come into existence at the moment of their conception, they will never pass out of existence.

Therefore, humans have an endless future. And that future also will be equally real to God.

From his eternal perspective outside of time, God sees and creates all the moments of our endless future. They are both real—actual—from his perspective, and they are infinite in number. Being both actual and infinite, the moments of our future also would be an actual infinity.

From the viewpoint of a classical Christian theist, eternalism implies the existence of an actual infinity of future moments, giving such theists reason to challenge Craig’s first premise (that an actual infinity can’t exist).

However, this post is only examining his second premise, so let’s consider the other option we need to look at.

The Presentist Option

If only the present exists, is it true that a beginningless series of events in time entails an actually infinite number of things?

No. At least not an actual infinity of real things.

The reason is that, on the presentist view, only one moment of time exists. No past moments exist, and no future moments exist.

It doesn’t matter how many events took place in the past, because those events are no longer real. As soon as a new moment arrived, all the events taking place in the previous moment evaporated and are no longer actual.

Therefore, it doesn’t matter how many past events there have been—it could be a finite number or an infinite number—because they have all ceased to be actual. The only actual events are those occurring in the present.

So, if presentism is true, the second premise of Craig’s argument is false if applied to concrete, real things like events. A beginningless series of events in time does not entail an actually infinite number of such things because those things are no longer actual.

For a collection of things to be actually infinite, they all have to be actual from some perspective. On eternalism, that can happen, because all the moments of time are actual from the eternal perspective outside of time.

But it can’t happen on presentism, because this view holds that, from the ultimate perspective, only one moment is real, and one is a finite number. This view entails that no actual infinity of moments in time exists, because only one moment of time is actual.

This is why Aristotle could believe that the world did have an infinite history. Even though he thought an actually infinite number of things couldn’t exist at the same time, history didn’t present that problem, because one moment passed out of existence when another came into it, so the total number of moments was always finite.

The Counting Argument

In the Blackwell Companion to Natural Theology, Craig and coauthor James Sinclair respond to this issue with two lines of thought.

The first is based on counting, and their reasoning (omitting examples for brevity) goes like this:

[W]e may take it as a datum that the presentist can accurately count things that have existed but no longer exist. . . .

The nonexistence of such things or events is no hindrance to their being enumerated. . . .

So in a beginningless series of past events of equal duration, the number of past events must be infinite, for it is larger than any natural number. . . .

[I]f we consider all the events in an infinite temporal regress of events, they constitute an actual infinite.

It’s true that a presentist can count things that have existed but no longer exist (e.g., the number of days that have elapsed so far this year)—and their nonexistence doesn’t prevent this counting (just look at a calendar!).

The problem comes in the third statement, because it can be understood in more than one way.

In terms of what is real on the presentist view, the number of past events is not infinite, because no past events exist. That’s a key point of presentism.

If you want to talk about an infinite number of past events, you have to shift from speaking of events that do exist to those that have existed, and those aren’t the same thing.

Yes, on presentism, we could speak of an infinite collection of events that were real but aren’t anymore. And that’s the point: They aren’t real.

This points to a second way of reading the statement when Craig and Sinclair speak of “the number of past events.”

If we are talking about the number of events, then we’re no longer talking about the events themselves. Instead, we’re talking about a number, which raises a question.

What Are Numbers?

Mathematicians and philosophers have a variety of views about what numbers are. Some classify them as “abstract objects” that exist independent of the mind. Others think of them as mental constructs of some kind. There are many variations on these views.

Whatever the case may be, Craig doesn’t see infinite numbers themselves being a problem.

In his talks and writings, he has frequently said that he doesn’t have a problem with the mathematics of infinity—that modern mathematical concepts dealing with the infinite are fine and useful as concepts. Thus, the infinite set of natural numbers (0, 1, 2, 3 . . . ) is a useful concept.

Craig doesn’t reject the idea that the set of natural numbers is actually infinite. It’s just not the kind of actual infinity that causes a problem for him because numbers aren’t concrete objects in the real world.

So, actual infinities of the numerical order are fine, in which case it’s fine if the number of past events is actually infinite. It’s an actual infinity of events themselves that he says can’t be part of the real world.

And on presentism, they’re not. Past events would have to be understood in some other way. They might be abstract objects, like many mathematicians hold numbers to be. Or they might be purely mental concepts at this point, as others regard numbers.

Whatever is the case, on presentism they do not exist in the real world. And so, whatever kind of infinity a beginningless universe would involve, it doesn’t violate the principle that—while actual infinities may exist in an abstract way, as in mathematics—they don’t exist in the real world.

Back to the Future

There is another way of illustrating the problem with the argument from counting, and it involves considering the number of future events.

If the universe can’t have a beginningless past because an infinite set of non-real past events can’t exist, then we also can’t have an endless future, because that entails an infinite set of non-real future events.

The argument simply involves shifting from events that used to be real to those that will be real.

If God gives people endless life, then the number of days that we will experience in the future is infinite. As the hymn says about heaven,

When we’ve been there ten thousand years,

Bright shining as the sun,

We’ve no less days to sing God’s praise,

Than when we first begun.

As Craig and Sinclair acknowledge:

It might rightly be pointed out that on presentism there are no future events and so no series of future events. Therefore, the number of future events is simply zero. . . . [O]n presentism, the past is as unreal as the future and, therefore, the number of past events could, with equal justification, be said to be zero. It might be said that at least there have been past events, and so they can be numbered. But by the same token there will be future events, so why can they not be numbered? Accordingly, one might be tempted to say that in an endless future there will be an actually infinite number of events, just as in a beginningless past there have been an actually infinite number of events.

So, why should an infinite number of future events be considered more permissible for a presentist than an infinite number of past ones?

Possible vs. Actual Infinity

Craig and Sinclair’s response involves the difference between an actual infinity (where an unlimited number of elements exist simultaneously) and a potential infinity (where an unlimited number of elements don’t exist simultaneously). They write:

[T]here never will be an actually infinite number of [future] events since it is impossible to count to infinity. The only sense in which there will be an infinite number of events is that the series of events will go toward infinity as a limit. But that is the concept of a potential infinite, not an actual infinite. Here the objectivity of temporal becoming makes itself felt. For as a result of the arrow of time, the series of events later than any arbitrarily selected past event is properly to be regarded as potentially infinite, that is to say, finite but indefinitely increasing toward infinity as a limit.

This reasoning is mistaken. It is false to say that “the series of events later than any arbitrarily selected past event is . . . finite but indefinitely increasing toward infinity as a limit.”

No. If you arbitrarily select any event in time and consider the sequence of later events, they do not “indefinitely increase toward infinity.” They are always infinite.

Consider January 1, 1900. On the Christian view, how many days of endless life will there be after that? An infinite number.

Consider January 1, 2000. How many days are to come? Again, an infinite number.

Consider January 1, 2100. How many days follow? Still an infinite number.

As the hymn says, “We’ve no less days to sing God’s praise than when we first begun!”

What Craig and Sinclair are thinking of is the fact that, if you pick a date and go any arbitrary distance into the future, your destination will still be a finite number of days from your starting point.

Thus, the number of days that has elapsed between the start and finish of your journey grows toward infinity but never gets there, making this span of days a potential rather than actual infinity.

But it does not follow—and is simply wrong—that the complete set of future days is only potentially infinite. To show this, just give each day a number: Today is 0, tomorrow is 1, the next day is 2, and so on. We can thus map the set of future days onto the set of natural numbers, which is actually rather than potentially infinite.

Take any day you like, and on the Christian view the quantity of days that will be after it is identical to the quantity of natural numbers.

The quantity of days that will be—like the quantity of natural numbers—does not grow. This quantity just is.

Unless you say—contrary to the teaching of the Christian faith—that the number of future days is finite and God won’t give us endless life, then there is an actual infinity of future days.

And if a presentist wants to affirm an actual infinity of currently-not-real days that will be, he must allow the possibility of an actual infinity of currently-not-real days that have been.

Conclusion

In summary, Craig’s second premise was:

2) A beginningless series of events in time entails an actually infinite number of things.

Whether this is true will depend on one’s view of time and the status of non-real things.

On eternalism, a beginningless series of events in time would involve an actually infinite number of things, for all these moments exist from God’s perspective outside of time. But so would the actually infinite number of future days that God promises us, giving the eternalist reason to reject the idea that an actual infinity cannot exist in the real world.

On presentism, a beginningless series of events in time would not involve an actual infinity of events existing in the real world, because presentism holds that the past does not exist.

Such a series of events might result in an actual infinity of (past) non-existent days, but so would the actual infinity of (future) non-existent days. And if a Christian allows one set of non-existent days, the other must be allowed as well.

The fact that the past days are countable is irrelevant, because so are the future days.

And it is simply false to say that the days that will be are only potentially infinite. They’re not. Right now, the number of days that will be is actually infinite, the same way the set of natural numbers is actually infinite.

Based on what we’ve seen here, presentism does not exclude an infinite past any more than it does an infinite future.

God created the universe a finite time ago, but there’s a question of whether we can prove this by reason alone.

Defenders of the Kalaam cosmological argument often claim that the universe cannot have an infinite history because “traversing an infinite” is impossible.

In his book Reasonable Faith (pp. 120-124), William Lane Craig puts the argument this way

1. The series of events in time is a collection formed by adding one member after another

2. A collection formed by adding one member after another cannot be actually infinite

3. Therefore, the series of events in time cannot be actually infinite.

The second premise of this argument is the one that deals with “traversing an infinite.” Craig writes:

Sometimes this problem is described as the impossibility of traversing the infinite.

Still a third way of describing it is saying that you can’t form infinity “by successive addition.”

Whatever expression you prefer, each of these expressions refer to the intuition people commonly have about infinity—that “you can’t get there from here.”

Where Is “Here”?

If you can’t get to infinity from here, where is “here”?

However you want to phrase the problem—getting there from here, traversing an infinite, or successive addition, this is a question that needs to be answered.

Let’s take another look at the second premise:

2. A collection formed by adding one member after another cannot be actually infinite

What does it mean to “form” a collection by adding one member after another?

Perhaps the most natural way to take this would be to form such a collection from nothing. That is, you start with zero elements in the collection (or maybe one element) and then successively add one new member after another.

And it’s quite true that, if you form a collection this way, you will never arrive at an infinite number of members. No matter how many elements you add to the collection, one at a time, the collection will always have a finite number of elements.

This can be seen through a simple counting exercise. If you start with 0 and then keep adding +1, you’ll get the standard number line:

0, 1, 2, 3, 4, 5, 6, 7 . . .

But no matter how many times you add +1, the resulting number will always be finite—just one unit larger than the previous finite number.

However, there is a problem . . .

The First-and-Last Fallacy

As I’ve discussed elsewhere, any string of natural numbers that has both a first and a last element is—by definition—finite.

Any time you specify a first natural number and a last natural number, the space between them is limited.

It thus would be fallacious reasoning to envision an infinite timeline with both first and last elements.

Yet it is very easy to let the idea of an infinite past having a beginning somewhere “infinitely far back” unintentionally sneak back into discussions of the Kalaam argument.

It can easily happen without people being aware of it, and often our language is to blame:

• The natural sense of the word “traverse” suggests going from one point to another, suggesting both a beginning point and an end point.
• So does the idea of “forming” an infinite collection. If we imagine forming a collection, we naturally envision starting with nothing (a collection with no members) and then adding things to it.
• And if we think of getting to infinity “from here,” we naturally think of a starting point in the finite realm (“here”) and an end point (“infinity”).

Without at all meaning to, it’s thus very easy to fall into the trap of subconsciously supposing both a starting point and an ending point in a supposedly infinite history.

This happens often enough that I’ve called it the First-and-Last Fallacy.

Taking No Beginning Seriously

In Reasonable Faith, Craig denies that this is how his argument should be understood. He writes:

Mackie and Sobel object that this sort of argument illicitly presupposes an infinitely distant starting point in the past and then pronounces it impossible to travel from that point to today. But if the past is infinite, they say, then there would be no starting point whatever, not even an infinitely distant one. Nevertheless, from any given point in the past, there is only a finite distance to the present, which is easily “traversed.” But in fact no proponent of the kalam argument of whom I am aware has assumed that there was an infinitely distant starting point in the past. The fact that there is no beginning at all, not even an infinitely distant one, seems only to make the problem worse, not better (boldface added).

Craig thus wishes us to understand his argument not as forming an infinite collection of past historical moments from an infinitely distant starting point—i.e., from a beginning.

It’s good that he is clear on this, because otherwise his second premise would commit the First-and-Last Fallacy.

But does this really make things worse rather than better?

It would seem not.

Formed from What?

If we are not to envision a collection being “formed” from nothing by successive addition, then it must obviously be formed from something. Namely, it must be formed from another, already existing collection.

For example, suppose I have a complete run of my favorite comic book, The Legion of Super-Heroes. Let’s say that, as of the current month, it consists of issue #1 to issue #236.

Then, next month, issue #237 comes out, so I purchase it and add it to my collection. I now have a new, larger collection that was “formed” by adding one new member to my previous collection.

Now let’s apply that to the situation of an infinite history. Suppose that the current moment—“now”—is the last element of an infinite collection of previous moments (with no beginning moment).

How was this collection formed?

Obviously, it was formed from a previous collection that included all of the past moments except the current one.

Let’s give these things some names:

• Let P be the collection of all the past moments
• Let 1 represent the current moment
• And let E represent the collection of all the moments that have ever existed

With those terms in place, it’s clear that:

P + 1 = E

We thus can form one collection (E) from another collection (P) by adding a member to it.

But Can It Be Infinite?

Now we come to Craig’s second premise, which said that you can’t form an actually infinite collection by adding one member after another.

If you imagine forming the collection from nothing—and thus commit the First-and-Last Fallacy—then this is true.

But it’s not true if you avoid the fallacy and imagine forming an actually infinite collection from a previous collection by adding to it.

The previous collection just needs to be actually infinite as well. If P is an actually infinite collection and you add 1 to it, E will be actually infinite as well.

And this is what we find in the case of an infinite past. Let us envision an infinite past as the set of all negative numbers, ending in the present, “0” moment:

. . . -7, -6, -5, -4, -3, -2, -1, 0.

The set of all the numbers below 0 is infinite, but so is the set of all numbers below -1, all the numbers below -2, and so on. Each of these collections is actually infinite, and so we can form a new, actually infinite set by taking one of them and adding a new member to it.

Understood this way, Craig’s second premise is simply false. You can form an actually infinite collection by adding new members to an actually infinite collection—which is what we would have in the case of a universe with an infinite past, one that really does not have a starting point.

Conclusion

What we make of Craig’s argument will depend on how we take its second premise.

Taken in what may be the most natural way (forming an infinite collection from nothing—or from any finite amount—by successive addition), will result in the argument committing the First-and-Last Fallacy.

But if we take it in the less obvious way (forming an infinite collection by adding to an already infinite collection), then the second premise is simply false.

There may be other grounds—other arguments—by which one might try to show that the universe cannot have an infinite past.

But the argument from “successive addition,” “traversing an infinite,” or “getting there from here” does not work.

Depending on how you interpret it, the argument either commits a fallacy or uses a false premise.