Category: Philosophy

“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.

This page collects articles I have been writing dealing with the Kalaam cosmological argument for God’s existence. It is expected to grow over time as I write more on the subject.

Put simply, the argument goes:

1. Whatever has a beginning has a cause.
2. The universe has a beginning.
3. Therefore, the universe has a cause.

The cause of the universe can be meaningfully described as God. Therefore, God exists.

This argument is both valid and sound–that is, it uses a valid logical form and its premises are true, so its conclusion is true.

Despite this, many of the arguments used to support it are not successful. In particular, many of the philosophical arguments used for its second premise are flawed.

Here are articles in which I explore why.

General Considerations:

• Using the Kalaam Argument Correctly
• Omnipotence and Infinite History
• Catholic Teaching and the Kalaam Argument

The Big Bang and Science

• Science and the Kalaam Argument: Some Words of Caution

Philosophical Arguments:

The First Argument (No Actual Infinities):

• Can an Actual Infinity Exist?
• Presentism and Infinite History
• Checking Out of Hilbert’s Hotel

The Second Argument (No Infinity by Successive Addition):

• Traversing an Infinite?
• What’s Wrong with the Countdown Paradox?

More Recent Arguments:

• Grim Reapers, Paradoxes, and Infinite History
• The Kalaam Cosmological Argument and the First-and-Last Fallacy

Related:

• 3 Views of Time and Eternity
• Does Only the Present Exist?
• Are the Past and the Future Real?

Sometimes defenders of the Kalaam cosmological argument defend its second premise (i.e., that the world couldn’t have an infinite past) by proposing a paradox involving counting.

The line of reasoning goes something like this:

A. Suppose that the universe has an infinite history (the kind of history you’d need to do an infinite countdown).

B. Suppose that a person has been counting down the infinite set of negative numbers (. . . -3, -2, -1) for all eternity, and they finish today, so today’s number is 0. It took them an infinite amount of time to reach 0 in the present.

C. Now suppose that we go back in time to yesterday. How much time was there before yesterday? Also an infinite amount of time! Given that, they could have counted down the infinite set of negative numbers so that they reached 0 yesterday instead of today!

D. So, we have a paradox: If the person had been counting down the negative numbers for all eternity, they could have finished today—or yesterday—or on any other day in the past, since there was always an infinite number of days before that.

E. There needs to be a sufficient reason why they stop on the day they did.

The Kalaam defender then challenges the Kalaam skeptic to name the sufficient reason, and if he’s not convinced by the answer, he rejects Step A of the argument—the idea that the universe has an infinite history—since there doesn’t seem to be anything wrong with Steps B, C, D, or E.

What’s problematic about this line of reasoning?

Arbitrary Labels

To see what the answer is, we need to think about the arbitrariness of the labels involved in the countdown.

In Part B, the Kalaam defender chose to use the set of negative numbers, but he could have chosen something else.

For example, he could have chosen the digits of the irrational number pi (3.14159 . . . ) in reverse order (. . . 9, 5, 1, 4, 1, 3), in which case today’s number would be 3.

Or he could have used the Golden Ratio and chosen the digits of the irrational number phi (1.61803 . . . ) and reversed them, in which case today’s number would be 1.

Or he could have picked anything else, such as an infinitely long string of random numbers—or random words—or random symbols.

Any string will do for an infinite count of the past—as long as it’s an infinitely long string.

The point we learn from this is that the labels we apply to particular days are arbitrary. It depends entirely on what labels we choose. We can pick any labels we want and use them for any set of days we want.

Forward Counts

To underscore this point, let’s consider counts that go forward in time rather than backwards.

For example, we could choose the set of natural numbers (0, 1, 2, 3 . . . ), assigning 0 to today, 1 to tomorrow, 2 to the day after that, and so on.

Or we could use the digits of pi, in which case today would be 3, tomorrow 1, the day after that 4, etc.

Or the digits of phi, so today would be 1, tomorrow 6, the day after that 1, etc.

Or we could use something else—such as an infinite string of random numbers, words, or symbols.

We can pick whatever labels for a set of days, beginning with today, that we want!

A Count-Up Paradox

Now consider the following line of reasoning:

A*. Suppose that the universe has an infinite future (the kind of future you’d need to do an infinite count going forwards).

B*. Suppose that a person starts counting the infinite set of natural numbers (0, 1, 2, 3 . . . ) today, so that today’s number is 0, tomorrow’s is 1, the next day is 2, etc.

C*. Now suppose that we go forward in time to tomorrow. How much time is there left in the future of the universe? Also an infinite amount of time! Given that, the person could start their count of the infinite set of whole numbers so that they begin with 0 tomorrow instead of today!

D*. So, we have a paradox: If the person counts the set of whole numbers for all eternity, they could have started today—or tomorrow—or on any other day in the future, since there will always be an infinite number of days after that.

E*. There needs to be a sufficient reason why they start on the day they do.

If we’re challenged to name the sufficient reason why the person starts counting on the day they do, what will our answer be?

Mine would be, “Because that’s how you set up the thought experiment! You made this determination in Step B*. You could have chosen to start the count on any day you wanted (today, tomorrow, yesterday—or any other day), and you chose the set of numbers that would be used to label these days. Your choices are the sufficient reason for why the count starts and why it labels the days the way it does.”

Turn About Is Fair Play

And this is the answer to the original line of reasoning we presented. The same logic is present in A-E that is present in A*-E*, so the answer is the same.

The reason that the original countdown stopped today, which was labelled 0, is because those were the choices made in Step B. The person setting up the thought experiment chose that the countdown stop today, and he chose that it would stop with 0.

Once again, it is the choices that the person made that determine when the count stops and what it stops on.

There is only a “paradox” here if you lose sight of the fact that these choices were made and demand a sufficient reason over and above them.

To say—in the first case—“I know I made these choices in Step B, but I want a reason over and above that to explain why the countdown doesn’t stop on another day” is the same as saying—in the second case—“I know I made these choices in Step B*, but I want a reason over and above that to explain why the count doesn’t start on another day.”

No such reasons are needed. The choices made in Step B are sufficient to explain why the countdown works the way it does, just as the choices made in Step B* are sufficient to explain why the count-up works the way it does.

So, like a lot of paradoxes, the “countdown paradox” has a perfectly obvious solution once you think about it.

God as the Decider

Now let’s apply this to the question of whether God could have created the universe with an infinite past. In this case, we’re doing a thought experiment where God is the one making the choices.

A**. Suppose that God creates a universe with an infinite past (the kind you need for an infinite countdown).

B**. Suppose that–within this timeline–God creates a person (or angel, or computer, or whatever) that counts down the negative numbers so that he finishes today, and today’s number is 0.

Why didn’t the person stop counting on some other day or with some other number? Because that’s not what God chose. He chose to have it happen this way, with the person counting the number -2 two days ago, the number -1 one day ago, and the number 0 today.

Could he have have done it differently? Absolutely! God could have made different choices!

In fact–to go beyond what we’ve stated thus far–God may have created other people doing just that.

C**. Suppose that God also created a second person who has been counting for all eternity such that he ended yesterday with the number 0.

D**. Suppose that God further created a third person who has been counting for all eternity such that he ended two days ago with the number 0.

These are also possible, and we can modify our thought experiment such that God creates any number of people we like, finishing an infinite count on any day we like, with any number (or word or symbol) we like.

In each case, it is God’s choice that is the sufficient reason why the person finished when he did and with what he did.

The situation is parallel to the following:

A***. Suppose that God creates a universe with an infinite future (the kind you need to do an infinite count going forward).

B***. Suppose that–within this timeline–God creates a person who starts an infinite count today, beginning with the number 0.

As before, we can include any number of counters we want:

C***. Suppose that God also creates a second person who begins counting tomorrow, starting with the number 0.

D***. Suppose that God further creates a third person who begins counting the day after tomorrow, starting with the number 0.

As before, we can modify our thought experiment to include any number of counters we want, they can start on any day we want, and they can start with whatever number (or word or symbol) that we want.

Yet in these scenarios, it is God’s choices that determine who is created, when they start counting, and how the count works. These choices are the only reasons we need to explain what is happening.

If there is no unsolvable paradox preventing the scenarios described in A***-D***, then there is no unsolvable paradox preventing the scenarios described in A**-D**–or in any of the previous scenarios we’ve covered.

There just is no problem with the idea of a person doing an infinite countdown ending today–any more than there is with the idea of a person beginning an infinite countdown today.

The Kalaam cosmological argument for God’s existence has been around for centuries.

It hasn’t been that popular historically, with authors such as St. Thomas Aquinas pointing out problems with it. However, it has been popular in recent years, due principally to the advocacy of William Lane Craig.

I’ve written and spoken about it on a number of occasions, and I won’t review all that here, but I’d like to point out some problems with some recent defenses that involve a logical fallacy.

Stating the Argument Briefly

One way of putting the Kalaam argument is:

1. Everything with a beginning has a cause.
2. The universe has a beginning.
3. Therefore, the universe has a cause.
4. The cause of the universe, by definition, is God.
5. Therefore, God exists.

In this article, the premise we’re interested in is line 2—that the universe has a beginning.

This has been argued both on scientific and philosophical grounds.

Modern Big Bang cosmology is consistent with the universe having a beginning, though—as Fr. George LeMaitre (the father of Big Bang cosmology) pointed out—it does not conclusively prove it.

The results of science are always provisional, so if you want a conclusive proof, you’ll need to demonstrate it philosophically and rely on logic rather than observation.

Consequently, defenders of the Kalaam argument have sought to prove by logic alone that the universe had a beginning.

Finding a Logical Contradiction?

From a Christian perspective, the question we need to ask is whether God could have created the world with an infinite history.

We know from Scripture that he did not. He created the universe at a specific point in the past, so it has a finite history.

The question is whether he could have created it with an infinite history if he chose. Is a universe with an infinite history something God would be able to bring about by his divine omnipotence?

Omnipotence allows God to create anything as long as it does not involve a logical contradiction—that is, as long as the terms involved do not contradict each other.

“Four-sided triangle” and “square circle” both involve logical contradictions (as does “stone too heavy for an omnipotent being to lift”), so God can’t create those as the terms involved are logical gibberish and don’t refer to conceptually possible entities. They’re just word salad.

This is St. Thomas’s point when he says that, “everything that does not imply a contradiction in terms, is numbered amongst those possible things, in respect of which God is called omnipotent: whereas whatever implies contradiction does not come within the scope of divine omnipotence, because it cannot have the aspect of possibility” (ST I:25:3; cf. SCG 2:25).

Therefore, if you want to say that God can’t create a universe with an infinite history, you’ll need to show that this concept involves a logical contradiction. If it doesn’t, then it’s something God can create.

The burden of proof thus falls to the Kalaam advocate to show that “universe with an infinite history” is self-contradictory.

Traversing the Infinite

A common strategy for doing this involves what is sometimes called “traversing an actual infinite.” The basic idea is this:

1. Suppose that you start counting 1, 2, 3 . . .
2. No matter how high you count, you will never arrive at infinity, because there is always a next number you can say.
3. Therefore, the universe could not start infinitely far back in time, because you couldn’t start counting and cover an infinite series of moments to get up to the present. You thus can’t traverse an actually infinite series.

All of this is true.

It’s also irrelevant.

Notice what the argument says: “The universe could not start infinitely far back in time” and then traverse an infinity of time to the present.

The argument is assuming that the universe had a beginning—namely, one infinitely far back in time.

But that would not apply to a universe with an infinite history.

Understanding the Infinite

To see why, we need to remember the nature of the infinite. Derived from Latin roots, the word means “no” (in-) “limit” (finis). Something is infinite if it is lacking a limit or end.

For an ordered series, this can happen in one of three ways, as illustrated by the number line:

1. The set of positive natural numbers has a limit at the beginning (i.e., the number 1) but no final limit. It just keeps going {1, 2, 3 . . .}.
2. The set of negative natural numbers as a limit at the end (i.e., the number -1), but no starting limit, so it goes {. . . -3, -2, -1}.
3. The set of all natural numbers (which also includes 0) has neither a beginning limit nor an ending limit, so it goes {. . . -3, -2, -1, 0, 1, 2, 3 . . .}.

Each of these three sets is infinite because they lack either a beginning limit, and ending limit, or both.

However, if an ordered series has both a beginning and an ending limit, then it isn’t infinite. Instead, it’s finite. Consider the sequence {6, 7, 8, 9}. This series is finite because it has both a beginning limit (6) and an ending limit (9). It is thus limited to the numbers between 6 and 9, making it finite (i.e., limited).

What’s Wrong with Traversing an Infinite?

With this in mind, we can see what’s wrong with the “Traversing an Infinite” argument, above.

It assumes that the universe had a beginning “infinitely far back in time” and then traversed an infinite series of moments to arrive at the present. In making these assumptions, it assumes both a beginning limit and a final limit. It therefore—by definition—describes a finite series.

If {H} is the set of moments in the universe’s history, then there can’t be both a first moment and a last (most recent) moment without {H} being a finite set.

The argument thus tells us nothing more than that a number can’t be both finite (limited) and infinite (unlimited).

We already knew that.

But it’s irrelevant to the issue of a universe with an infinite history.

We can agree that the universe’s history ends in the present, but if it had a beginning—however far back in time—then its history is, by definition, finite.

If a universe’s history ends in the present, it has a final limit. Therefore, for its history to be truly infinite, it must not have a beginning limit. Therefore, it has no beginning. It’s just always been there, with no first moment.

All this is pointed out by Aquinas, in summary form, when he writes that God could create a history containing an infinite number of previous revolutions of the sun, for “if the world had been always, there would be no first revolution. Wherefore there would be no passing through them, because this always requires two extremes” (SCG 2:38:11).

Therefore, you wouldn’t be traversing an infinity. When you traverse a distance, it must have both a starting and an ending point, making it finite. For it to be infinite into the past, it must have no beginning point. In that case, the universe would have always been approaching the present, without any moment that this trek began.

The First-and-Last Fallacy

It’s therefore a logical mistake—or fallacy—to apply a series with both a beginning and an end to the case of a universe with an infinite history.

Because this move involves there being both a first element and a last element in a supposedly infinite series, I will call it the “First-and-Last Fallacy.” Stated more formally:

The First-and-Last Fallacy occurs if and only if a person envisions a supposedly infinite series as having both a first and a last element.

A series can be infinite if it has only a first element, like the series {1, 2, 3 . . .}. Or it can be infinite if it has only a last element, like the series {. . . -3, -2, -1}. Or it can be infinite if it has neither a first and last element, like the series {. . . -3, -2, -1, 0, 1, 2, 3 . . .}.

But it can’t have both a first and last element and be infinite.

Unfortunately, the “Traversing an Infinite” argument isn’t the only one that commits the First-and-Last Fallacy.

Some recent philosophical defenses of the Kalaam argument also commit this fallacy. They are, in essence, alternate versions of the “Traversing an Infinite” argument presented in other guises.

The Grim Reaper Paradox

Alexander Pruss and Robert Koons have proposed what is known as the Grim Reaper Paradox, which can be formulated different ways. One way goes like this:

1. There is a man named Fred, who is alive at 12:00 noon.
2. However, in the next hour, he must face an infinite series of Grim Reapers with orders to kill him if he is not already dead.
3. The final Grim Reaper will kill him at 1:00 p.m. if he is still alive then.
4. But half an hour earlier, at 12:30, another Grim Reaper would kill him if he was still alive then.
5. And a quarter hour before that, at 12:15, another Grim Reaper would kill him if he was still alive then.
6. The rest of the infinite series of Grim Reapers are set to strike in increasingly shorter intervals, as we work our way back to 12:00 noon, when Fred was alive.

The question is: Which Grim Reaper kills Fred?

• He will definitely be dead by 1:00 p.m., because the final Grim Reaper will kill him.
• But that one shouldn’t be the answer, because he should have been killed by the 12:30 p.m. Grim Reaper.
• And that one shouldn’t have had to kill him, either, because of the 12:15 p.m. Grim Reaper, and so on.
• It thus looks like we can’t identify which Grim Reaper kills him, because there’s always a prior Grim Reaper who should have done the job.

Framed this way, the problem is essentially the same as asking “What’s the first moment of time after 12:00 noon?”—because that’s when Fred will encounter the first Grim Reaper and get killed.

But if time is infinitely divisible, there is no “first” moment after noon, because you can always specify a prior moment. If you say “1 second” after noon, you then could say “1/2 second earlier,” then “1/4 second earlier,” then “1/8 second earlier,” and so on.

We encounter the same problem on the number line. If you consider all the decimal numbers between 0 and 1, there is no limit to them, and there is no “first number after 0.” If you propose 0.1, you’ll have to face 0.01, then 0.001, then 0.0001, and so on. There’s always a number closer to 0 and thus there is no “first” number after 0.

That’s what’s required by an infinite series.

This reveals what’s wrong with the Grim Reaper Paradox. If the number of Grim Reapers Fred will encounter really is infinite, with the final one striking at 1:00 p.m., then there is no first Reaper. There can’t be if the series is truly infinite, just like there can be no “first element in the infinite series {. . . -3, -2, -1}” or a “first number after negative infinity.”

These are logically impossible entities.

The New Grim Reaper Paradox

Some have reformulated the Grim Reaper Paradox so that it doesn’t involve time compression between noon and 1 p.m.

For example, you could have the Grim Reapers strike on New Year’s Day. If Fred is alive on New Year’s Day 2021, a Grim Reaper will kill him. But there also was a Grim Reaper set to strike on New Year’s Day in every prior year, going back an infinite number of years in history.

The same issue then results. There is no first Reaper that kills Fred.

And that’s the problem: The thought experiment is proposing an entity that can’t exist, because it involves a logical contradiction.

There can’t be a first Grim Reaper in an infinite series that has a final limit any more than there can be a first number in the series {. . . -3, -2, -1}. That’s a contradiction in terms.

It’s therefore a logical impossibility.

The terms involved in the thought experiment entail a logical contradiction. The question “Who is the first Grim Reaper that Fred encounters in an infinite series of Reapers that ends at 1:00 p.m. on New Year’s Day 2021?” is the logical equivalent of “What is the first decimal number after 0?” or “How many angles does a 4-sided triangle have?”

These are all just word salad.

And the particular form of word salad in the Grim Reaper Paradox involves there being a first and a last Grim Reaper in a supposedly infinite series of them. It thus commits the First-and-Last Fallacy.

The Eternal Society Paradox

A similar, more recent paradox is proposed by Wade Tisthammer. It can be framed like this:

1. Suppose there is a society that stretches infinitely far into the past (i.e., an eternal society).
2. Every year, this society flips a coin.
3. If the coin comes up “heads” (H) and it’s the first time it’s ever come up heads, they do a special chant to commemorate it being the first time heads has come up.
4. If the coin comes up “tails” (T), they do nothing.
5. Every combination of heads and tails is possible.
6. One possible combination is that in every prior year of their infinite history, the coin has come up tails, until the most recent year, in which it comes up heads, so the series looks like {. . . T, T, T, T, T, T, H}. In this case, they get together and do their special chant to commemorate it being the first time heads has come up.
7. Another possibility is that it comes up heads every year in their infinite history, so the series looks like {. . . H, H, H, H, H, H, H}.
8. In that case, when did they do the special chant to commemorate the first time heads came up? It can’t be in any of these years, because heads had occurred in every prior year.

It’s worth pointing out that an infinite series of heads is only possible. It’s not guaranteed, and so the paradox might not arise in actual history. If God has a rule that prevents logical contradictions from arising—a logical equivalent of the Chronology Protection Conjecture—then one could argue that the series {. . . H, H, H, H, H, H, H} will never arise.

However, we don’t need to go that route, because, after what we’ve seen, the solution is straightforward: The Eternal Society Paradox is presupposing a logical contradiction.

It’s asking “What is the first heads in an infinite series of head flips that has a final limit?”

If an infinite series has a final element, then it can’t have a first element—by definition.

Once again, we’re dealing with the equivalent of “What’s the first decimal number after 0?” or “How many angles does a four-sided triangle have?” The thought experiment presupposes an entity that involves a logical contradiction and thus cannot exist.

It presupposes a first and a last element to a supposedly infinite series, so the Eternal Society Paradox commits the First-and-Last Fallacy.

Physicalizing the Infinite

We’ve looked at three philosophical arguments that the universe must have a finite history:

• Traversing an Infinite
• The Grim Reaper Paradox
• The Eternal Society Paradox

Each one involves the First-and-Last Fallacy, though they present it in different physical terms. They thus physicalize it in different ways and correspondingly ask different questions:

• Traversing an Infinite asks us to envision a starting point to the universe’s infinite history and then asks how we could cross that by traversing an infinite number of moments to get to the present. The answer to this question is that we couldn’t. You can’t cross an infinite series by successive addition. However, this is irrelevant because—if the universe has an infinite history terminating in the present—then there was no first moment when the journey began. The universe has just always been there, getting progressively closer to the present.

• The Grim Reaper Paradox asks us to envision an infinite series of Grim Reapers that end at a certain moment (say, 1:00 p.m. on New Year’s Day 2021) who are all ordered to kill Fred if he hasn’t already been killed. It then asks who is the first Grim Reaper to encounter Fred and kill him. Answer: None of them. There is no first Reaper in an infinite series that terminates at a certain moment. Such a Reaper is a logically impossible entity—as is “an infinite series of Reapers with a first and last element.” Such a self-contradictory series cannot exist, and thus cannot harm Fred. He’s either alive or not, regardless of the word salad entity that’s supposed to confront him.

If a series of Reapers is truly infinite into the past, then Fred has simply never lived, except for the window between his birth and when the next Reaper was timed to strike.

• The Eternal Society Paradox invites us to consider an infinite series of coin tosses that all turn up heads and that ends in the present. It then asks us when the Eternal Society did its special chant to commemorate the first time the coin came up heads. Answer: Never. There was no “first time” the coin came up heads. “First heads flip in an infinite series of head flips ending in the present” is a logically impossible entity. Therefore, in physical terms, the Eternal Society would never have done their chant to commemorate the first heads flip, because there wasn’t one.

Each of these thought experiments presupposes an entity that involves a contradiction in terms—a series that has both a first element and a last element and yet is supposed to be infinite.

They may then go on to ask questions about an element in this series, which also involves a logical contradiction (e.g., “What’s the first element in an infinite series that has a final limit?”).

But this is a nonsensical question, just like “What’s the first number after negative infinity?”, “What’s the last number before positive infinity?” or “If there were a four-sided triangle, what size would it be?”

In view of how easy it is to fall into the First-and-Last Fallacy, advocates of the Kalaam argument who wish to support it with philosophical proofs need to carefully examine their arguments to see whether they are envisioning a scenario that presupposes both a first and a last element in a supposedly infinite series.