Defenders: Excursus on Natural Theology (Part 9): 1st Philosophical Argument for the Beginning of the Universe

Summary

Defenders Podcast – Excursus on Natural Theology (Part 9):

1st Philosophical Argument for the Beginning of the Universe

Host: Dr. William Lane Craig
Date: June 8, 2022

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Episode Overview

This episode dives deeply into the second premise of the Kalam Cosmological Argument: "The universe began to exist." Dr. Craig explains and defends the philosophical case for the universe having a beginning, focusing chiefly on the distinction between potential and actual infinities, the implications of actual infinities, and the famous Hilbert’s Hotel paradox. Throughout, Dr. Craig fields student questions to clarify core concepts and potential objections.

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Main Discussion Points & Insights

1. Introduction to the Second Premise

• [00:14] Dr. Craig recaps the Kalam argument’s first premise dealt with previously, then focuses on the much-debated second premise:

  “Today we want to turn to the second premise of that argument, which is that the universe began to exist. This is obviously the more controversial of the two premises.” ([00:14])

• He plans to explore both philosophical and scientific support but identifies philosophy as the primary support, with science as confirmation.

2. Al Ghazali’s Argument & The Infinite Past

• Drawing from 12th-century Muslim theologian Al Ghazali:
  • If the universe never began, there must have been an actually infinite number of past events.
  • But, Al Ghazali argues, actual infinities cannot exist in reality.

3. Distinction: Potential vs. Actual Infinity

• Potential Infinite: Never completed; a process that goes on indefinitely.
  • “Potentially infinite... always finite at any point in time, but they're growing toward infinity as a limit.” ([03:40])
  • Symbolized by the lemniscate ("lazy 8," ∞); used as a limit concept in calculus.
• Actual Infinite: A completed, static collection with an infinite number of items.
  • “An actual infinite is an infinite which is, as it were, complete... involves an actually infinite number of things.” ([04:40])
  • Symbolized by the Hebrew letter Aleph (ℵ), notably Aleph-null (ℵ₀) in set theory.

4. Set Theory and Arithmetic with Infinity

• Set theory allows mathematicians to discuss actually infinite sets consistently—but only on paper, using specific rules.
• Transfinite arithmetic: With aleph numbers, addition and multiplication are defined and allowed (e.g., ℵ₀ + ℵ₀ = ℵ₀), but subtraction and division are prohibited to avoid contradictions.

  “You can do transfinite arithmetic using these numbers... What is interesting... you can't do inverse operations like subtraction and division with them, because then you get self-contradictions.” ([09:46])

5. Infinity and Real World Absurdities

• Using the example of odd/even infinite sets:

  “So what Al Ghazali would say is that while you can talk about actually infinite collections and do these mathematics on paper, it's not something that can exist in the real world because it will involve these sorts of counterintuitive absurdities.” ([07:13])

6. Modern Mathematics & Criticisms

• Set theory’s consistency is based on rules/axioms—it does not prove that actual infinites can exist in reality.

  “Modern set theory shows that if you adopt certain axioms and rules, then you can talk about actually infinite collections in a consistent way. But... it does absolutely nothing to show that such mathematical entities really exist...” ([17:13])

• The distinction between mathematical abstraction and physical reality is paramount.

7. God's Infinity

• God's infinity is qualitative, not quantitative.

  “The infinity of God is not the infinity of a collection that is made up of an infinite number of data, definite and discrete parts... it's more as it were, a qualitative infinite, not a quantitative infinite.” ([21:28])

8. Hilbert’s Hotel: The Core Illustration

• Hilbert’s Hotel is used to illustrate the absurdities that arise from the existence of an actual infinite in reality.
  • An infinite hotel, fully occupied, can still accommodate new guests (by shifting everyone one room over), even an infinite number of new guests.
  • Subtraction from infinite sets results in paradoxical outcomes: removing identical-sized subsets leads to different results (sometimes infinite, sometimes finite).

  • “[Hilbert’s Hotel] would have to have a sign outside, no vacancy. Guests welcome. Can such a hotel exist in reality?” ([27:16])

  • “So you subtract identical quantities from identical quantities and you get non identical results, which is absurd.” ([29:44])

9. Critiques & Responses

• Objection: “Absurdities only arise in Hilbert’s Hotel because of ‘moving things around’, while time’s past events can’t be rearranged.”
  • Craig refutes: The paradox is conceptual and functions with mental elimination, not physical rearrangement.

  • “...the number of past days in the history of the universe, it's easy to just mentally annihilate every other day... There would still be all the even numbered days which is the same number.” ([33:19])

• Objection: “That’s just how infinity works.”
  • Craig’s reply: “Of course this is what would happen if an actually infinite number of things could exist. But the question is, is such a hotel really possible? And I think that these illustrations show that no such a thing is not really possible. It is metaphysically absurd.” ([34:13])

10. Summary of the Argument

• Dr. Craig summarizes Al Ghazali’s first philosophical argument:

  1. An actual infinite cannot exist.
  2. An infinite temporal regress of events is an actual infinite.
  3. Therefore, an infinite temporal regress of events cannot exist. ([34:52])

• Conclusion: The universe must have had a beginning.

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Notable Quotes

• Infinity as a Conceptual Limit vs. Actual Number:

  "Aleph null is a number. It is a number. It's a quantity. It's the number of members in this set. The lemniscate, or the potential infinite, isn't a number... It's an ideal limit concept." — Dr. Craig ([08:48])

• Mathematical vs. Physical Existence:

  “If Ghazali is right, this universe of discourse may be regarded simply as a fictional realm, rather like the world of Sherlock Holmes in the Arthur Conan Doyle novels, not something that exists in the real world.” — Dr. Craig ([17:37])

• On God's Infinity:

  “The infinity of God is not a quantitative concept... It's more as it were, a qualitative infinite, not a quantitative infinite.” — Dr. Craig ([21:28])

• Hilbert’s Hotel Absurdity:

  “You subtract identical quantities from identical quantities and you get non identical results, which is absurd.” — Dr. Craig ([29:44])

• On Critics Who ‘Bite the Bullet’:

  “The only thing that the critic can do at this point is just bite the bullet and say... that Hilbert’s hotel is not absurd. Yeah, that's right. That's the way it would be... But... the question is, is such a hotel really possible?... It is metaphysically absurd.” — Dr. Craig ([34:13])

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Key Timestamps

• [00:14] — Introduction to the second premise (“the universe began to exist”)
• [03:40] — Explanation of potential vs. actual infinity
• [07:13] — Odd/even number paradox and real-world absurdities
• [09:46] — Arithmetic with Aleph numbers and limitations
• [13:00] — Difference between set-theoretical abstractions and reality
• [21:28] — Infinity of God: qualitative vs. quantitative
• [25:20] — Hilbert’s Hotel and its paradoxes
• [29:44] — Subtraction paradox in Hilbert’s Hotel
• [33:19] — Addressing criticism regarding past events and Hilbert’s Hotel
• [34:52] — Summary of Al Ghazali’s argument

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Memorable Moments

• Dr. Craig delights in students grasping subtle distinctions, especially when they distinguish between limit concepts and numerical values for different kinds of infinity.
• The lively give-and-take on whether “actual infinity” is a contradiction, the permissibility of various arithmetic operations in transfinite math, and the theological implications regarding God's nature.
• The use of Hilbert’s Hotel as both an entertaining and intellectually forceful illustration of the issues at stake.

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Conclusion

Dr. Craig forcefully argues—via philosophical reasoning, illustration, and pedagogical Q&A—that an actually infinite number of past events is metaphysically impossible. Thus, the universe must have had a beginning—a crucial step in the Kalam Cosmological Argument. The episode is rich with clarifying interplay between philosophy, mathematics, and theology, offering a highly accessible yet rigorous exploration for listeners at any level.

Transcript

[Dr. William Lane Craig]

Welcome to Defenders, the teaching class of

[Student 1]

Dr. William Lane Craig.

[Dr. William Lane Craig]

Today, an excursus on Natural Theology, Part 9.

[Student 1]

For more resources from Dr. Craig, go

[Dr. William Lane Craig]

to reasonablefaith.org Last week we began talking about the Kalam cosmological argument, and I offered a defense of the first premise. Today we want to turn to the second premise of that argument, which is that the universe began to exist. This is obviously the more controversial of the two premises. It's fairly obvious, I think, that if the universe began to exist, then the universe has a cause of its existence. But it's by no means obvious that the universe began to exist. So I want to examine both philosophical arguments and scientific evidence in support of this second premise. And if you were to ask me what the relationship is between these two, I would say that for me, at least, the first line of defense for this second premise is the philosophical arguments. And I see the scientific evidence as simply a confirmation empirically of a conclusion already established on the basis of philosophical arguments. So I will often speak of the support for this premise in terms of philosophical arguments and scientific confirmation. Now let's look at the first philosophical argument. Al Ghazali, the 12th century Muslim theologian whom we've taken as our springboard for examining this argument, argued that if the universe never began to exist, then there has been an infinite number of past events prior to today. But, he argued, an infinite number of things cannot exist, and therefore it follows that there cannot have been an infinite past. Now, Al Ghazali recognized that a potentially infinite number of things could exist, but what he denied was that an actually infinite number of things could exist. So it's important that we understand this absolutely crucial distinction between the potential infinite and the actual infinite. When we say that something is potentially infinite, we mean that something is indefinite but progressing toward infinity. As an ideal limit, which is never reached, you never actually arrive at infinity. Infinity is simply a limit concept which you approximate toward. For example, take any finite distance. You could divide that distance in half and then into fourths, then into eighths, then into ,16, then into 30 seconds ad infinitum. But you would never reach an infinitief division. The number of divisions is potentially infinite in that you can go on dividing endlessly, but you never arrive at infinity. You would never have an actually infinite number of divisions or of parts. And the symbol for this kind of infinity is the lemniscate, or the lazy eight. And this is the type of infinity that is used in calculus in mathematics, where you have infinite limits. Now, by contrast with that the actual infinite is an infinite which is, as it were, complete. The number of items in the collection is not growing toward infinity. It is infinite. It is complete and static and involves an actually infinite number of things. This type of infinity is symbolized by the Hebrew letter Aleph and is used in set things theory. In set theory, mathematicians talk about sets like the set of natural numbers, which have an actually infinite number of members in the set. The collection is not growing toward infinity as a limit. It is infinite. They're an actually infinite number of natural numbers in. In this set. So Aleph is a number. If you were to ask what is the number of elements in the set of natural numbers? The answer would be aleph null, a subscript zero that is the number of members in the set of natural numbers. Now, technically speaking, what defines a collection or a set as actually infinite is that it has a proper part which has the same number of members as the whole collection. It has a proper part with the same number of members as the whole collection. So, for example, think about this. The number of odd numbers is the same as the number of all the natural numbers, namely Aleph null. It's exactly the same. There are just as many odd numbers as there are natural numbers, even though the natural numbers includes not only the odd numbers, but an infinite number of even numbers as well. So, technically speaking, the definition for an actual infinite is that it's a collection that has a proper part with the same number of members in it and as the whole collection. And what Al Ghazali is claiming is that while potentially infinite, collections can exist, that is to say, collections that are always finite at any point in time, but they're growing toward infinity as a limit. There cannot be a collection that is actually infinite that has an actually infinite number of members in it. Now, before we proceed, I. I think we need to pause and ask if there are any comprehension type questions about this distinction, because it is so fundamental any understanding question about the difference between actual and potential infinity. And don't be embarrassed to say I don't understand this. I need more explanation. Okay. All right. Do you want to. Let's get the microphone to him. And can you state the reason for your perplexity or what needs more explanation? You indicated that the odd numbers would be equal to the actual number of numbers, so it seems like that to me. It's just half of the odd and even numbers. Yes, and I'm glad that you feel that way, because this is precisely what will engender various absurd situations when you translate this out of the mathematical realm into the real world of people and sticks and rocks and eggs and things like that, you get extremely bizarre results precisely because of this. So what Al Ghazali would say is that while you can talk about actually infinite collections and do these mathematics on paper, it's not something that can exist in the real world because it will involve these sorts of counterintuitive absurdities.

[Student 2]

Cindy, just to clarify for me, so we're saying. So the odd numbers. If you have an infinite number of odd numbers and an infinite number of total numbers, then they're both infinite, therefore the same number.

[Dr. William Lane Craig]

That's correct.

[Student 2]

On the potential infinite. That would be a concept where you could imagine something indefinite.

[Dr. William Lane Craig]

Yes.

[Student 2]

And therefore it's potentially. But it's a concept of infinity.

[Dr. William Lane Craig]

Yes.

[Student 2]

Rather than an actual number of items.

[Dr. William Lane Craig]

Yes. Very good, Cindy. I am so excited. Notice. Steve, can you get the mic back from her? Notice the distinction that Cindy saw. Aleph null is a number. It is a number. It's a quantity. It's the number of members in this set. The lemniscate, or the potential infant isn't a number. It's not a number. It's a limit. It's an ideal limit concept, but it's not a number. And that's important to understand. Okay, very good. Yes. Jim, those folks in Houston said people are smart. They were right.

[Student 3]

I know. The lazy 8 is not a number, as you said, and you can't do math on it.

[Dr. William Lane Craig]

Well, you can do calculus with it. It's used in calculus.

[Student 3]

Okay, it's a. Right. But you can't, like, multiply it by two because then it's equal to itself. But in what way is the Aleph, if I said that right, a number? Because you can't do math on that either.

[Dr. William Lane Craig]

Well, actually you can, Jim. This is the interesting thing with these Alephs, because there's really more than one of them. Remember I said the number of natural numbers is Aleph null, but there are more real numbers than there are natural numbers. And so you begin to get a whole series of these Alephs that have subscripts. Aleph null, aleph1, aleph2, and it goes to infinity. There's actually an infinite number of these infinites. I mean, this is where it just becomes completely beyond the human mind to comprehend. And you can do mathematical operations with these numbers. This is called transfinite arithmetic. For example, what is Aleph null plus Aleph null? Well, the answer is Aleph null. So you can do transfinite arithmetic using these numbers. You can do Multiplication and you can do addition and you can do exponentiation like Aleph null to the second power, for example. So this is a number that can be manipulated in arithmetic in this way. What is interesting, and this will become significant when we talk about whether actual infinites can really exist, is that you can't do inverse operations like subtraction and division with them, because then you get self contradictions. And so it's stipulated, it's part of the rules, that all you can do are these positive operations like addition and multiplication. And you can't do subtraction and division. That's prohibited. So one over Aleph isn't zero. Right? You can't do division with these sorts of things. Yes. Charles, in the back you had said

[Student 3]

that you can have an infinite number of Alephs. So would that be Aleph to the Aleph?

[Dr. William Lane Craig]

That's a good question. I think that the number of Alephs is Aleph null because they're enumerated by the natural number 0, 1, 2, 3, 4. So if you have them subscripted with the natural numbers, the number of Alephs would be Aleph null.

[Student 1]

Okay, so it's not a.

[Student 3]

Is it an actual infinite of Alephs

[Student 1]

or is it a potential infinite?

[Dr. William Lane Craig]

Yeah, right. Yeah. Okay. One more comment from Steve.

[Student 1]

Isn't the ratio of Aleph n divided by Aleph n +1 always zero by definition? Because you can't put them in a one to one relationship.

[Dr. William Lane Craig]

You're saying an Aleph null divided by a finite natural number N. If you

[Student 1]

take aleph n, say Aleph 2.

[Dr. William Lane Craig]

Oh, okay.

[Student 1]

And divide that by Aleph 3, the answer is always zero.

[Dr. William Lane Craig]

Well, you can't do those kind of inverse operations, Steve. That is prohibited. You're trying to do division with these and you can't do that.

[Student 1]

But I think you, I think you can prove that it's zero. I think that's known. I could be wrong.

[Dr. William Lane Craig]

Yes. So far as I know, you can't do those kinds of inverse operations like dividing one Aleph by another.

[Student 1]

This is speaking to the size of it. Next one is always.

[Dr. William Lane Craig]

Well, it is always. That's right. These are different sizes of infinites. That is correct. The Aleph one is a larger collection than Aleph null. So in that sense it has members in it that the other one doesn't have. All right. Now, Al Ghazali, as I say, has no problem with the idea of merely potential infinites. These are just ideal limits. But he argued that if an actually infinite number of things could exist, then various absurdities would Result. And if we're to avoid these absurdities, we have to deny that an actually infinite number of things can exist. And that would imply that the number of past events in the history of the universe therefore cannot be actually infinite. It must be finite. And therefore the universe cannot be beginningless. The universe must have begun to exist. Now, it's very frequently alleged. Well, let me pause there and ask, is there any question of a comprehension type nature about Al Ghazali's argument? Yes, Kurt. What are the absurdities? I mean. Well, I'll go into those in a moment. I just want. Yes, of course, of course. I just wanted you to make sure we're all tracking together. And I don't get ahead of you, but you understand the basic argument, namely, it's absurd that an actually infinite number of things could exist, but a beginning, less past, would involve an actually infinite number of things, namely past events. So the past can't be actually infinite. It must be finite and therefore have a beginning. That's the basic argument that I want to make sure we all get.

[Student 1]

John, I'm one of those that doesn't quite understand to me the better way to understand what the term actual infinite means, what you're trying to do with saying actual is not infinite. So if you. Rather than saying infinite, you could say finite. So seems like what you're saying, it's like not infinite means it is finite. So it seems like we're starting with a contradiction of terms to begin with.

[Dr. William Lane Craig]

Well, that's why it's important, John, to make sure we're on the same page with the definitions.

[Student 1]

I'm not there yet.

[Dr. William Lane Craig]

Okay. When mathematicians talk about an actual infinite, as I say, they don't mean what you just said, that this is some kind of finite thing or contradiction. It means that the collection is complete. It's not growing toward infinity as a limit. There's a real infinite number of things in that collection, and that's the force of the word action.

[Student 1]

And that's a contradiction in terms of.

[Dr. William Lane Craig]

Well, now, that's very interesting. Is it? I'm going to speak to that in a minute. If you follow the rules and the conventions laid down by set theory, you won't run into any contradictions. It isn't logically contradictory if you follow the rules and obey the axioms and conventions. But there's the rub, and I'll say something about that in a minute.

[Student 4]

Yes, thank you, Dr. Craig. So you're saying that you can add it and multiply it, but you can't reverse it? Yes, okay, I'm not sure I follow. Follow how? You can. Not sure how I follow how you can add it, multiply it, but then you can't reverse the process.

[Dr. William Lane Craig]

No, no. What are you.

[Student 4]

Back to the original?

[Dr. William Lane Craig]

Okay, and what is the question?

[Student 4]

Well, you said that you could add and multiply the actual infinity.

[Dr. William Lane Craig]

Right.

[Student 4]

But you couldn't divide it or subtract it.

[Dr. William Lane Craig]

Right.

[Student 4]

So you can't reverse it.

[Dr. William Lane Craig]

Right.

[Student 4]

I don't understand how you can't go back to the original.

[Dr. William Lane Craig]

Okay, I'm sorry, I missed that last page. So you're saying how do you take it back to the original?

[Student 4]

Right.

[Dr. William Lane Craig]

You can't. Mathematically. Mathematically you can't. Now, the difficulty is, I think that if this is something that really exists, like it's a bunch of eggs or coins or people, you could. And I think that illustrates what we're saying, is that while this works on paper, if you obey the rules, there's no reason to think that those sort of rules hold in reality, and you're going to get into difficulties. And I have yet, as Kurt says, to illustrate these. All right, now let me just say that it is very frequently alleged that Al Ghazali's sort of argument is. Is invalidated by modern mathematics. In modern set theory, as I've said, the use of actually infinite sets is commonplace. The number of members in the set of natural numbers is actually infinite, not just potentially infinite. And many people have inferred that given the coherence of infinite set theory in mathematics, that this sort of argument is just a non starter. But is that really the case? Modern set theory shows that if you adopt certain axioms and rules, then you can talk about actually infinite collections in a consistent way without contradicting yourself. As I said in response to John's question, now all this does is succeed in setting up a certain universe of discourse for talking consistently about actual infinites. But it does absolutely nothing to show that such mathematical entities really exist or that an actually infinite number of things can really exist. If Ghazali is right, this universe of discourse may be regarded simply as a fictional realm, rather like the world of Sherlock Holmes in the Arthur Conan Doyle novels, not something that exists in the real world. Is there any question about that point? Yes, Bruce has a comment.

[Student 4]

Couldn't you also criticize the criticism by

[Dr. William Lane Craig]

applying Goodell's incompleteness theorem that you. In any mathematical system, you're going to assume something is true in order for it to work, and you can't prove the assumption true within the. I don't think that that is relevant to the concern that we're raising here. I think it would mean you couldn't prove the consistency of infinite set theory. But we're not trying to do that. So I don't think that result is is pertinent to the question we're raising here.

[Student 4]

I was just trying to criticize the criticism.

[Dr. William Lane Craig]

Yeah, yeah. Okay. Now I want to restrict this to comprehension type questions at this point. Yes.

[Student 3]

Just so I can understand where we're at so far.

[Dr. William Lane Craig]

Yes.

[Student 3]

The adding and multiplying of the Alephs is possible because they're both infinite. But taking from it would obviously make it a part of an infinite which doesn't exist. Which proves the point point that having every odd number equals the same amount of every other numbers also can't exist. So no actual infinity actually exists, except for possibly God. So the only infinity, the real infinity, that never had a beginning, never has an end, that's always constant infinity is just God.

[Dr. William Lane Craig]

Okay, you've raised a number of questions there. The reason that you can't do these inverse operations in transfinite arithmetic is because you get self contradictory results. Let me give an example. Suppose you take the natural numbers and you subtract all the odd numbers. How many numbers are left over? Well, all the even numbers. Right? So infinity minus infinity is infinity. But suppose instead you subtract from the natural numbers all the numbers greater than than two. Now how many are left over? Well, three. So infinity minus infinity is three. In fact, you can get any answer to infinity minus infinity, from zero to infinity. So as I say, there is no well defined result for the equation infinity minus infinity equals blank. You can get any answer from zero to infinity. So you get self contradictory answers. So these operations are simply prohibited to the mathematician. Now, with respect to God, people will often ask this question, but isn't God infinite? And here I think it's very important to understand that the infinity of God is not a quantitative concept. God is not a mathematical quantity. So the infinity of God is not the infinity of a collection that is made up of an infinite number of data, definite and discrete parts. When theologians talk about God as infinite, it's more as it were, a qualitative infinite, not a quantitative infinite. That is to say, God is omnipotent, omniscient, morally perfect, eternal, necessary, all loving. And those aren't quantitative concepts. Indeed, in a sense there isn't any separate attribute of God called infinity. It's kind of just an umbrella term for all of his superlative attributes. If you were to take away in your mind. Omniscience, omnipotence, eternity, necessity, holiness. There wouldn't be any attribute left over called infinity that just is an umbrella term for all those superlatives, attributes that God possesses. So we shouldn't think of God's infinity as a quantitative concept. He doesn't involve an actually infinite number of definite and discrete pieces that go to make up his being. Alright, yes, okay, take another question.

[Student 5]

Sorry to ask another question just about God's infinity though.

[Dr. William Lane Craig]

Yes.

[Student 2]

Is it possible though that with God

[Student 5]

he may be able to understand and do calculations with an actual infinite.

[Dr. William Lane Craig]

So for example, might God have considered an actually infinite number of counterfactuals before creating this universe? Wow, okay, you're getting into very difficult issues of metaphysics now. What Kevin is raising is this old problem that we've encountered again and again, and that is, do abstract objects exist? Because propositions or counterfactuals would be examples of abstract objects. If there are abstract objects like mathematical objects, numbers, propositions, possible worlds, properties, then these are plausibly actually infinite. But I'm persuaded that these things don't exist and that therefore they do not contradict Al Ghazali's statement that there cannot be an actually infinite number things. So the anti realist isn't bothered by those sorts of counterexamples. In order for that to be an effective counter example to Al Ghazali, you would need a proof that Platonism is true. And there isn't any such proof. Platonism is just one alternative among many and it's not incumbent upon us. Okay, we really get into the deep weeds when we start talking about these things. All right, now the way Al Ghazali brings out the real impossibility of an actually infinite number of things is by imagining what it would be like if such a collection could exist and then drawing out the absurd consequences of it. Let me share with you one of my favorite illustrations called Hilbert's Hotel, which is the brainchild of the great German mathematician, the David Hilbert. Hilbert warms up by inviting us to imagine an ordinary hotel with a finite number of rooms. And let's suppose that the rooms are completely occupied. There is not a single vacant room throughout the entire hotel. Now suppose a new guest shows up at the front desk asking for a room. Sorry, the manager says all the rooms are occupied and the new guest has to be turned away. But now Hilbert imagines, let's suppose we've got a hotel with an infinite number of rooms. And let's suppose once again that the hotel is completely occupied. And we have to fully appreciate this fact. There is not a single vacancy in the entire infinite hotel. Every room has a flesh and blood person in it. Now, suppose a new guest shows up at the front desk asking for a room. No problem, says the manager. And he moves the guest that was in room one into room two. He takes the guest that was in room two and puts him in room three. He takes the guest that was in room three and puts him in room four out to infinity. As a result of these transpositions, room one now becomes vacant and the new guest is easily accommodated. And yet before he arrived, all the rooms were already full. It gets even worse. Now, Hilbert says, let's imagine that an infinite number of new guests shows up at the front desk asking for rooms. No problem, no problem, says the manager. And he moves the person who was in room one into room two. The person who was in room two into room four, the person who was in room three into room six. He puts each person into the room number double his own. Now, since any number multiplied by two is always an even number, that means all of the guests wind up in the even numbered rooms and all of the odd numbered rooms become vacant. And the infinity of new guests gratefully checks in. And yet before they arrived, all of the rooms were already full. As one student remarked to me, Hilbert's hotel, if it could exist, would have to have a sign outside, no vacancy. Guests welcome. Can such a hotel exist in reality? Since nothing hangs on the illustrations involving a hotel, this argument can be generalized to show that the existence of an actually infinite number of things is really absurd. Now, I hadn't planned on sharing further difficulties with Hilbert's hotel, but given that it's already come up, let me say that the German mathematician didn't even fully demonstrate the absurdity of this hotel because he never asked what would happen if people started checking out of the hotel. Let's suppose that the people in the odd numbered rooms check out 1, 2, 3, I mean 1, 3, 5, 7, and so forth. All the odd numbered guests check out how many guests are left? Well, all the even numbered guests, so an infinite number of guests are still left in the hotel, even though an equal number has already checked out and left the hotel. But now let's suppose instead that all of the guests in the rooms 3, 4, 5, 6, 7, out to infinity check out how many guests are left now. Well, if there's a room zero, just three are left. And yet the same number of guests checked out this time as when all of the Odd numbered guests left. So you subtract identical quantities from identical quantities and you get non identical results, which is absurd. Now someone might say, but you can't do inverse operations with mathematical quantities. Not on paper perhaps, but there's no way you can stop people from checking out of a real hotel. If you try to bar the door, they'll go out the windows. So this illustrates the absurdity of the real existence of an actually infinite number of things. Now, sometimes students will react to Hilbert's hotel by saying that these absurdities result because the concept of infinity is just beyond us and we don't understand it. But that reaction is mistaken and naive. As I said, infinite set theory is a highly developed and well understood branch of modern mathematics. These absurdities result because we do understand the nature of the actual infinite. Hilbert was a smart man and he well knew how to illustrate the bizarre consequences of the existence of an actually infinite number of things. Any discussion of that argument against the existence of an actually infinite number of things?

[Student 6]

I just wanted to say that I taught this argument at Mount Vernon Presbyterian School to a bunch of high schoolers and we got to this philosophical understanding of actual infinite. And I used the example that you gave in Lee Strobel's case for creator with marbles. If you have an infinite number of marbles and you want to give another person an infinite number of marbles, you could do it different ways and you would get absurd results. And I just wanted to say that they really enjoyed talking about it and they understood it. So anyone who says they can't understand these things, like high schoolers can really get into this kind of stuff. So they just really enjoyed it. And anyway, I just wanted to say.

[Dr. William Lane Craig]

Okay, thanks for the encouraging word. Yes, Cody?

[Student 5]

Yeah, you stated that infinity in mathematics, you know, the reason they're not allowed to do all these sort of subtraction, etc. Is because you get contradictions. But they say as long as you don't do that, it's not contradictory or anything. And it reminds me of a point because, you know, Wes Morriston likes to bring up the point that, like say, the contradictions arise when you move people around, whereas the past, which is what you're trying to argue is finite, isn't something you can't move around past dates like you can people in a hotel or coins or marbles.

[Dr. William Lane Craig]

I've never understood why someone thinks that. That is a good objection. We're obviously not talking with regard to Hilbert's hotel about a real hotel that's built out of bricks and Wood and has people trying to walk down infinite hallways to get out the door. I mean, it's a conceptual thought experience experiment. You imagine the hotel with all the people in the rooms and then as it were in thought, just eliminate all the people in the odd numbered rooms, just vaporize them or something. And then you've got all the people left in the even numbered rooms. You don't want to get into difficulties about physically moving them about and so forth. And similarly, with respect to the number of past events, if you imagine the number of past days in the history of the universe, it's easy to just mentally annihilate every other day or all the odd numbered days and ask how many are left over. And the answer is obvious. There would still be all the even numbered days which is the same number. So it seems to me that that kind of objection just fails to reckon with the nature of a third thought experiment which isn't based upon real physical movements and operations.

[Student 5]

And the other thing is, because it seems a lot, a lot of the major, the more knee jerk reaction that some people will have, mostly atheists, they'll say something like, well, okay, but that's, there's no absurdity. That's just what happens when you have infinity. That's just the way infinity works and there's no problem to it.

[Dr. William Lane Craig]

Okay, thank you for saying that. Because that's the segue to the next point, which is really the only thing that the critic can do at this point is just bite the bullet and say, as Cody said, that Hilbert's hotel is not absurd. Yeah, that's right. That's the way it would be. And sometimes they'll justify this by saying that if an actual infinite could exist, then such situations are exactly what we should expect. But again, I don't think this is an adequate response. Hilbert would of course agree that if an infinite hotel could exist, then the situation that he has imagined is what we would expect. Otherwise it wouldn't be a good illustration. Right? So of course this is what would happen if an actually infinite number of things could exist. But the question is, is such a hotel really possible? And I think that these illustrations show that no such a thing is not really possible. It is metaphysically absurd. So I think that Al Ghazali's first argument is a good one. It shows that the number of past events must be finite and that therefore the universe had a beginning. And we can summarize this argument as follows. Premise 1, an actual infinite cannot exist. 2, an infinite temporal regress of events is an actual infinite 3. Therefore, an infinite temporal regress of events cannot exist. Next time we will look then at the second independent argument that Al Ghazali offers for the beginning of the universe and the finitude of the past. But let's bow now to close our call with a benediction. The grace of the Lord Jesus Christ and the love of God and the fellowship of the Holy Spirit be with us all. Amen. The copyright for the content of this recording is held by Dr. William Lane Craig. For more go to reasonablefaith.org.