I. Introduction

The modern philosopher and prolific debater William Lane Craig is well-known for renewing and updating an old argument for the existence of God, called the Kalaam Cosmological Argument. One premise of the KCA, as amended by WLC, is the following:

(INF)Anactual infinityis impossible.

To support (INF), WLC usually offers a few supporting reasons. Chief among these are the following:

(1) Referring to Hilbert's famous thought experiment of the Grand Hotel, which involves a hypothetical hotel with a countably infinite number of rooms. Craig describes various resulting properties of the Hotel as "absurd" or "against reason" or "contradictory". The chief complaint is against the following counter-intuitive result: the Hotel can simultaneously have every room occupied--but can accommodate additional guests.

(2) Referring to transfinite arithmetic, Craig states that when we consider "infinity minus infinity", we arrive at contradiction. For example, the integers without the even numbers leaves the odd numbers, an infinite set, hence "infinity minus infinity" in this case is infinity; the integers without any number greater than 5 or less than 1 leaves only the set {1,2,3,4,5}, a finite set, hence "infinity minus infinity" in this case is 5. Craig argues that these results are contradictory, and that if there was an actual infinity, such operations must be able to be performed, and thus we would arrive at a contradiction.

These objections are essentially the same: infinite sets result in absurdity or contradiction. The objection in (1) is to the possibility of "adding" a set to an infinite set without increasing the infinite set's cardinality; the objection in (2) is to the inability to unambiguously determine the result of "subtracting" a set from an infinite set.

Now, WLC does not claim that these examples involvelogicalcontradictions. Rather, he claims that these examples show that "infinity" is necessarily constrained to the realm of imagination or ideas.

NOTE ON NOTATION: Although technically there are infinitely many "sizes" of infinity, within the context of this post I will use the word "infinity" to refer specifically to aleph-null, the cardinality of the set of natural numbers.

II. Problems

I don't see howanyof his examples forestall the existence of an actual infinity.

IIa. Hilbert's Grand Hotel

That Hilbert's Grand Hotel can always accommodate (countably) more guests may seem counter-intuitive, but this result in and of itself does not entail that an actual Grand Hotel could not exist. There might be other limitations that would prevent the construction of a Grand Hotel; it would require an infinite amount of energy/mass/etc.

But with regard to thecapacityof a Grand Hotel, I don't see how the ability to accommodate additional guests somehow renders anotherwise unobjectionableGrand Hotel contradictory.What rule of reason requires a careful thinker to reject the possibility of an actual Grand Hotel?Other than WLC's (and others') say-so, I don't see one.

IIb. Arithmetic with Infinity

Craig argues that because there is no sensible answer to the question, "What is infinity minus infinity?", there cannot be actual infinities. Here is where more rigorous formalization is needed.

When we talk about "infinity minus infinity", what we're actually talking about areset complements. Suppose that S is a subset of U. This means that every element in S is also in U. Then thecomplement of S in U, written U\S, is defined to bethe set of elements in U but not in S. If we consider the set N = {0, 1, 2, ..., k, k+1, ...}, then N\{0} = {1,2, ..., k, k+1, ...}. Call this set A; thus, A = N\{0} = {1, 2, ..., k, k+1, ...}. Then N\A = {0}.

So what is meant by "X minus Y"? Roughly speaking, it means "how many things are left over when you take away X many things from Y?" The labels "X" and "Y" refer to what are called cardinal numbers. Intuitively, cardinal numbers answer the question "how many?" Thecardinalityof a set S, written |S|, refers to the "number" of elements contained in S. In the example above, |N| = |A| = infinity, while |{0}| = |{1}| = ... = |{k}| = |{k+1}| = 1, etc.

In order to answer the question, "What is X minus Y?", first you find sets S, T such that: (1) |S| = X; (2) |T| = Y. Then X - Y is defined to be |S/T|. In order for this operation to be well-defined, your choice of S and T should not change the result of |S\T|. That is, if the operation is well-defined, then no matter which sets you use as S and T, the result |S/T| must be the same so long as |S| = X and |T| = Y.

This definition of "subtraction"--the cardinality of a set complement--is how Craig derives what he calls 'contradictions' or 'absurdities'. When Craig gives examples of what infinity "minus" infinity equals, he is using precisely this definition. For example, using the notation above, N - 2N = N\{0, 2, 4, ..., 2k, 2(k+1), ...} = {1, 3, 5, ...}, and thus "infinity minus infinity" equals infinity. But since |A| is infinity, "infinity minus infinity" must also equal |N\A| = |{0}| = 1. Thus infinity = 1, a contradiction. This shows thatsubtraction cannot be well-definedwhen both sets are infinite.

What does this mean?It means there is no unique solution to the following equation:

- = ___

But what contradiction does this entail?It makes it more difficult to figure out what |S\T| is, if we know that both S and T are infinite; we'd need to know the actual elements of S and T in order to determine |S\T|. But I don't see how the inability to determine |S/T| knowing only |S| and |T| results in absurdity, should an actual infinity exist.

I see no reason to conclude from Hilbert's Grand Hotel and from the ambiguity of subtraction on infinite numbers that an actual infinity cannot exist.III. Conclusion

Anyone who thinks such reasons exist, please present them here.

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