In this post I hope to sketch out the problem of actual infinities as is often utilized in the Kalam cosmological argument, the response that I am unable to respond to, and the basic question whose answer would settle the issue for me but I am unable to answer.
First, a quick note on mathematical symbols to clear things up for those who are not familiar with mathematical notation or the notation I sometimes made up because I did not know the proper way to express my idea. The main symbol I am going to use is the hebrew letter aleph (ℵ), which has come to represent an actual infinity in set theory. This is used instead of the more commonly known lemniscate (∞), which represents only a potential infinity or something without bound that would not be considered to include an infinite number of things. Also, I use the subscript 0 with the aleph to represent a certain kind of actual infinity, in this case it will usually be some subset of the natural numbers. This is only to be more specific as there are many larger infinite sets, a fact that is not relevant to my argument here. Moreover, I will use superscripts with aleph to specify the exact numbers included in the infinite set that the aleph stands for. This is only to make things clearer. I am not sure if that is proper notation. Hopefully the rest is pretty straightforward.
At first, arithmetic with transfinite cardinals seems fairly straightforward. It is simplest to imagine, the set of all natural numbers, as the infinite set under consideration. Cardinality is simply the property of any set that identifies the number of members in that set and is usually indicated by normal numbers used as cardinals. If I say I have 2 hands, then I could also say the set containing my hands has a cardinality of 2. This is different from ordinal numbers that indicate rank or position such as 1st, 2nd, and 3rd or nominal numbers that simply label something such as when we label athletes with numbers on their jerseys. One easy to understand property is that adding a new member to a set with infinite cardinality does nothing to change its cardinality. So taking the set of all natural numbers and giving it another 1 as a member does not make it any greater of an infinity.
In fact, adding an infinite number of members to the set does not affect the sets cardinality. So the union of the set of all natural numbers to the set of all natural numbers would produce a set with the same cardinality as the set of all natural numbers.
So addition does not lead to anything logically problematic, though it obviously is strange and difficult to grasp intuitively. However, this is not seen as a problem that prevents an actual infinity from being metaphysically inconceivable.
Actual Infinities and the Kalam Cosmological Argument
The problem for actual infinities steps in when one thinks of inverse operations such as subtraction. Again, things start out simple if you consider removing a single member of an infinite set because you are still left with an infinite set as would be expected by the rules of infinite set theory.
The source of controversy crops up when one considers removing an infinite number of members from an infinite set because it can result in a set with any cardinality between 0 and infinity.
This is exactly what apologists like William Lane Craig are trying to illustrate when they use examples like Hilbert’s Hotel. If this more formal take on it doesn’t make sense then I would recommend listening to William Lane Craig describe the problem using Hilbert’s Hotel. Anyway, one can illustrate this problem by thinking of removing all the odd numbers from the set of natural numbers, leaving one with an infinite set, namely, all the even numbers.
On the other hand, by removing all numbers greater than 3 from the set of all natural numbers, one is left with a set containing 3 members and therefore having a cardinality of three.
This could be repeated with different sets to achieve a set with any cardinality one wants. The problem then is that subtraction seems to be ill-defined and leads to the unacceptable and possibly illogical conclusion that removing an identical number of members from the same set can lead to different answers.
The proponent of the Kalam cosmological argument is going to argue that this proves the metaphysical impossibility of an actual infinite. This is because mathematicians can simply restrict inverse operations by fiat to prevent these illogical conclusions from following and then move forward in developing further theories. However, in the real world, nothing would prevent things from being removed from a set (though there are some persuasive arguments against this) and therefore such a fiat could not be imposed to prevent the illogical conclusions as is done in mathematics. Given the logical absurdity that could arise if an actual infinity existed in reality, it must be the case that actual infinites are metaphysically absurd and therefore do not exist in any possible world. From here the Kalam argument attempts to show that the past cannot be actually infinite, that space cannot be actually infinite, that an actually infinite number of universes cannot exist, and so on.
Objecting to the Kalam
The main objection I hear to this and the objection against which I cannot respond is that by simply taking into account the exact makeup of the infinite sets, than the metaphysical absurdities do not result. By keeping in mind that we are removing the infinite-set-of-all-even-numbers from the infinite-set-of-all-natural-numbers, it will always be the case that the result is the infinite-set-of-all-odd-numbers. It is only in ignoring the exact make-up of the infinite sets that absurdities result in performing inverse operations.
The issue that this all seems to boil down to is the relationship of different infinite sets of equal cardinality. The proponent (and possibly the objector) of the Kalam argument assumes that such sets are equal.
The controversial assumption is then whether this equality implies the ability to substitute sets of equally infinite cardinality for one another in an equation. If such substitution is allowed than the contradiction can be made undeniably explicit.
However, if the objector denies that equality between sets implies the possibility of substitution, then the logical problem is avoided and inverse operations become metaphysically possible.
So the most general way of stating the issue that is at the root of this whole argument, but that I am unable to know the answer to is this: can two infinite sets with equal cardinality be considered equal to each other such that substitution between equations is possible regardless of the exact makeup of either infinite set?
Implications for Cosmological Arguments
In my opinion, this is what the entire Kalam Cosmological Argument rests on. I don’t think anyone can seriously object to the first premise when it is properly stated, though I acknowledge that some people try. However, premise 2 and the conclusion that God must be the cause of the universe rest on this claim. For if actual infinities are not metaphysically impossible, the atheist can simply accept that fullness of the Kalam’s 3 premises and still maintain their atheism.
Atheists simply have to hold to the following: The universe, taken to be our current spacetime reality or any larger spacetime reality of which it is a part, has a cause. However we do not know that this cause must be personal. Instead we claim that minimally it is some type of mechanistic generator, that is constantly in a state of causal sufficiency for creating various kinds of universes and has therefore created an infinite number of them. We don’t know what this timeless and spaceless cause is and we may never know but it consistently explains everything we currently know about the universe without adding on the additional property of it being a person. We could even tack on that it is necessary in response to the Leibnizian cosmological argument. So God would work as an explanation as well but that would be ruled out by Ockham’s razor because the necessary mechanistic universe generator is the same in every respect to coherently explaining the existence, contingency, and fine-tuning of the universe but does so in a simpler way by not unnecessarily ascribing personhood to the cause.
Actual infinities seem to be one of the few core issues of contention in apologetic concerns surrounding cosmology. Unless it is shown that actual infinities are metaphysically absurd in some substantial way that is more than just being non-intuitive, atheists would seem to have no problem accepting the soundness of cosmological arguments. I have attempted to show that giving good reasons for the metaphysical absurdity of infinity boils down to whether or not inverse operations can be defined for infinite sets. This issue requires a mathematical background beyond what I currently have and so I would not presume that this a knock down objection in any sense. However, I hope this brings attention to what my research as an undergraduate philosopher and physicist has indicated as one of the (if not the) central issues for cosmological arguments in Christian apologetics.