In this brief paper I hope to sketch out the problem of actual infinities as is often
utilized in the kalam cosmological arguments, 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.
At first, arithmetic with transfinite cardinals seems fairly straightforward. It is
simplest to imagine ℕ, the set of all natural numbers, as the infinite set under
consideration. On obvious property is that adding a new member to a set with infinite
cardinality does nothing to change its cardinality
ℵ0 + 1 = ℵ0
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.
ℵ0 + ℵ0 = ℵ0
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.
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.
ℵ0 − 1 = ℵ0
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.
ℵ0 − ℵ0 = {∃𝑥: 0 ≤ 𝑥 ≤ ℵ0 }
One can illustrate this by thinking of removing all the odd numbers from the set of natural
numbers, leaving one with an infinite set.
ℵ01,2,3,… − ℵ01,3,5,… = ℵ0 2,4,6,…
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.
ℵ01,2,3,… − ℵ0 4,5,6 = 𝑆 1,2,3 = 3
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.
ℵ0 − ℵ0 ≠ ℵ0 − ℵ0
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.
However, in the real world, nothing would prevent things from being removed from a set
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.
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 no 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.
ℵ01,3,5,… = ℵ0 4,5,6,…
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.
ℵ01,2,3,… − ℵ0 4,5,6… ≠ ℵ0 2,4,6,…
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?
ℵ𝛼 = ℵ𝛼 ?
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 infinites are not metaphysically
impossible, the atheist can simply accept that fullness of the Kalam’s 3 premises and still
maintain their atheism. They simple 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 could
minimally claim that it is some mechanistic force or object, that is constantly in a state of
causal sufficiency for creating universes and has therefore created and infinite number of
them. We don’t know what this timeless and spaceless cause is and we may never know but
it consistenly explains everything we currently know about the universe without adding on
the additional property of 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 mechanistic universe
generator is the same in every respect to explaining the existence, contingency, and finetuning of the universe but does so in a simpler way by not ascribing personhood to the
cause.