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TMA | KCA 15 | IAINT | The Actual Infinite

Updated: Oct 1, 2021

Welcome to Tuesday Morning Apologetics (TMA). We’re talking about the Kalam Cosmological Argument (KCA):

1. Whatever begins to exist has a cause.

2. The universe began to exist.

3. Therefore, the universe has a cause.

Our focus is now on the second premise of the KCA which is supported by an argument called, “The Impossibility of an Actually Infinite Number of Things” (IAINT):

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.

First of all, I’d like to say that IAINT is one of the worst acronyms I’ve seen. And I was in the Navy and then in a nuclear industrial environment. So, I’ve seen a ton of acronyms and this one is bad. Reminds me of “I ain’t”. Maybe it’s short for, “I ain’t… a good argument.

Ha ha… heh.

Welp, we’ll see. That’s the point of this series, after all. To explore arguments for the existence of God and see if they’re any good.

The key premise in this argument is its first premise. Is it really true that “an actually infinite number of things cannot exist”? In order to really understand this premise we need to explain a couple things.

There are different kinds of infinites. The first one I'll mention is normally identified by a lemniscate (). This kind of infinite is like counting and never ending. It describes something that continuously progresses towards infinity but isn’t, itself, infinite. The second one is normally identified by aleph-null (ℵ0). This is different because it is talking about a complete set. There is no progression towards anything.(1)

For this argument, we are talking about the second kind of infinite. (ℵ0)

And when it is said that “an actually infinite number of things cannot exist”, it means being not some kind of potential. It needs to be actual and in order for it to be actual it must first be possible. And, just like infinites, there are different kinds of “possibles”. For our purposes we’re only going to talk about three.

Physical, Metaphysical and Logical

Things that are physically possible are things permitted by space-time, energy and the physical laws. Things that are metaphysically possible includes everything physically possible but also includes some things that are physically impossible. So, just because something is physically impossible, it doesn’t mean that it’s metaphysically impossible. And things that are logically possible includes all things physically and metaphysically possible but also things that are physically and metaphysically impossible. Anything that is logically impossible is simply impossible.

If it can be shown that something is metaphysically impossible then, necessarily, it is also physically impossible. And that’s what Dr Craig endeavors to do when he appeals to Hilbert’s Hotel. I have provided a few links below for you to see what Dr Craig means by Hilbert’s Hotel. It’s his favorite way to illustrate the absurdities produced by an actual infinite. Do these absurdities mean that an actual infinite number of things is plausibly metaphysically impossible?

Next week we’ll be exploring whether or not Hilbert’s Hotel provides us with sufficient warrant for believing the truth of the first premise of the IAINT argument. So, please be sure to look into those links or go watch one of Dr Craig’s debates. It comes up a lot.



  1. This video by Vsauce does a great job describing the infinites and I highly recommend watching it. How To Count Past Infinity


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