Here is a paper I wrote some time ago on the kalam cosmological argument. It's one of the most convincing arguments I've come across for the existence of God. The notation for the transfinite arithmetic is very elementary, since I was just beginning to learn the subject at the time. But I think the reasoning is still correct. But I would like to see more clarity in the connection between inverse operations and what is always possible in reality.

**Introduction**

The purpose of this paper is to show that the kalam cosmological argument is upheld despite arguments from some critics based on eternalism. The kalam cosmological argument has been vigorously opposed by highly respected philosophers of science such as Adolf Grunbaum. Many of these opponents have rejected the kalam type argument by taking an eternalist view on time. However, if the eternalist view on time is true, and if time does not have a beginning, then time can be thought of as an infinite set of finite durations of time. By looking at the metaphysical implications of transfinite arithmetic, I will argue that the existence of an infinite set of finite objects is metaphysically impossible, and hence the kalam argument is upheld on either an eternalist or presentist view of time.

This paper is structured to first give a brief overview of the two prevailing theories of time: presentism and eternalism. Then I will review the implications of these two views on time on the kalam argument. Finally, I will show that transfinite arithmetic implies that, if one adopts the eternalist view of time, an infinite regress of causes is impossible, and hence the kalam cosmological argument is upheld.

**Presentism and Eternalism**

A much more complete presentation of the history and definitions of presentism and eternalism is given in the Stanford Encyclopedia of Philosophy.

For this paper, it suffices to understand these two by asking the question, “are objects in the present all that exist, or do objects in the past and future exist just as much as objects in the present?” The presentist view of time is the most intuitive. It would answer the above question by saying that all that exists are objects in the present. Objects in the future do not exist but will exist, and objects in the past do not exist now but have existed. This is sometimes also referred to as the tensed theory of time, since the terms “will exist” and “have existed” are thought to not be reducible concepts.

In contrast, the eternalist views objects in the future and past as just as real as objects in the present. Time is viewed very much like the dimensions of space, in the sense that different objects can exist at different points in space at the same time. Similarly, future objects are different from present objects, but they both hold existence. Time exists as a dimension, and any sense of time passing is something that is only experienced by minds. This is sometimes viewed as the “tenseless” view of time because it does away with the verbs “will be” and “have been”. Everything simply exists at different points relative to each other in time.

**Two Versions of the Kalam Cosmological Argument**

The kalam cosmological argument is a particular kind of cosmological argument for the existence of God that argues for an absolute beginning to the universe based on the philosophical impossibility of an infinite amount of time to have passed. There are essentially two different approaches to the kalam cosmological argument, as put forth by Craig et. al.

Craig expresses the first of these two approaches as:

1) A collection formed by successive addition cannot be an actual infinite.

2) The temporal series of events is a collection formed by successive addition.

3) Therefore, the temporal series of events cannot be an actual infinite.

This is a very solid and straightforward argument. The first premise can be shown true through a version of Zeno's paradoxes where the intervals used are infinite rather than finite. A very solid argument is given in Craig (2009).

However, some philosophers such as Grunbaum and Craig have pointed out that this argument is only valid when one adopts a presentist view of time. Under the eternalist view, time is no different from any of the dimensions of space. On this view of time, there is no notion of temporal becoming, and time is not formed by successive addition. Instead, it exists as a completed dimension where each point of time in that dimension is just as real as the next. Eternalism undermines this particular version of the cosmological argument by denying that time is a temporal series of events, and hence it denies premise (2) of the argument. Craig defends his argument by attempting to show that temporal becoming is indispensable. However, since presentism and eternalism are still highly controversial topics in metaphysics, I propose to sidestep this difficulty by arguing for the second version of the kalam argument as put forth by Craig.

- An actual infinite cannot exist.
- An infinite temporal regress of events is an actual infinite.
- Therefore, an infinite temporal regress of events cannot exist.

Craig mentions several arguments in support of premise (1), particularly thought experiments. He also mentions the contradictions in trans-finite arithmetic, but does not go into detail explaining exactly what gives rise to the contradictions. He also seems to not recognize the power of this argument to carry the conclusion despite the challenge of eternalism. Craig writes, “From start to finish, the kalam cosmological argument is predicated upon the [presentist] theory of time.”

However, this argument here can be slightly recast to remove any notion of temporality. Instead, I argue that any infinite dimension or array is not metaphysically possible.

- An actual infinite set of finite objects cannot exist.
- Time without a beginning is an actual infinite set of finite objects.
- Therefore, time must have had a beginning.

It is without question that premise (2) is true by definition of an eternalist view of time without a beginning. All that remains then is to show that premise (1) is true.

**Trans-finite Arithmetic and Inverse Operations**

The notion of infinity has had a long history, beginning with Aristotle. Today, the most respected treatment of infinities is Cantorian theory, which was established by the mathematician Cantor in the late nineteenth century.

In transfinite arithmetic, it is important to distinguish between a potential and actual infinity. A potential infinity is some set that goes to infinity without bound, but does not achieve it. However, at any moment, a potential infinite is actually finite. In the context of time, a good example of a potential infinity is the time between now and the end of time. Presumably time will continue without end, so that at any point in time, there will be a finite amount of time between now and some point in the future. On the other hand, an actual infinity is some quantity that does achieve infinity. The most famous example of an actual infinity is the set of positive integers, called aleph zero. This set of numbers is already infinite in the sense that it has already achieved an infinite quantity, rather than going towards infinity.

The method that mathematicians have used since Bolzano to tell whether one set was equivalent in size to another is to see if the two sets can be put in a one-to-one correspondence with each other. An example of placing two sets in a one-to-one correspondence with each other is any function f(x). For example, the function f(x)=2x is able to place the set of positive integers into a one-to-one correspondence with the set of even integers. This led to the famous definition proposed by Dedekind in 1963 for the infinite: any set is infinite if a subset of that set can be put into one-to-one correspondence with the original set.

Addition with transfinite numbers and integers can be done easily by shifting the elements in one’s transfinite set and inserting the integers one would like to add. Imagine the transfinite number aleph zero:

0, 1, 2, 3.....

In this paper, I will use Ω to refer to aleph zero. We can write down the equation 3+Ω by placing a 0,1,2 in front of Ω in the series:

0’,1’,2’,0,1,2,3....

I have put a prime on the numbers from the 3 so that there will be no ambiguity. This set can be put into a one-to-one correspondence with Ω itself, and therefore we have found that 3+Ω=Ω. However, when one tries to add Ω+3:

0,1,2,3....0’,1’,2’

This is distinctly different from 3+Ω because we can see clearly that 2’ is the largest element, whereas there is no largest element from the set 3+Ω. Therefore, we cannot say that Ω+3=Ω in the same way we can say that 3+Ω=Ω, since we cannot put the elements in this series into a one-to-one correspondence with the elements in Ω. The point of this exercise is to note that addition is not commutative in transfinite arithmetic.

Note that nothing has been said that leads transfinite arithmetic into trouble thus far. However, when it comes to the inverse of these operations, the results will be ambiguous. For instance, consider any finite integer π, and a transfinite number ß We already know that the equation Ω+π=ß has an answer, and this operation can be inverted to find π. However, trying to solve π+Ω=ß by right hand subtraction will yield an infinite number of solutions for π. We know this is true because we already showed that 3+Ω=Ω, and in fact, any number π added to Ω will still yield Ω. Since any number solves the equation, the solutions are contradictory. A similar kind of logic in transfinite arithmetic will show that division also yields an infinite number of solutions, and hence is equally forbidden in transfinite arithmetic.

**Consequences**

If it is true that subtraction with transfinite numbers lead to contradictions, then it is clear that an actual infinite quantity cannot exist for any thing that can be subtracted in the real world. In reality, we can subtract lengths, times, or any other quantity that we like. Clearly, we cannot subtract qualities, but for any object that can be quantified, we can subtract. If subtraction is allowed in the real world, and if this subtraction will lead to a contradiction if the quantity is an actual infinite, then it must be true that it is metaphysically impossible for an actual infinite to exist. This part of the argument can be put more explicitly as follows.

1) Right handed subtraction in trans-finite arithmetic implies a contradiction.

2) Any operation that implies a contradiction is metaphysically impossible.

3)Therefore, right handed subtraction of an actual infinite is not metaphysically possible.

4) Right handed subtraction of any quantified object is always metaphysically possible.

5) Therefore, actual infinities are not metaphysically possible

This kind of reasoning has very far reaching consequences for our beliefs concerning space and time. Any dimension in space or time can be thought to be marked out evenly into finite segments. If actual infinities are metaphysically impossible, then it is impossible for any dimension to extend infinitely in any direction.

Spatially, it turns out that the current theory of space is that it is not infinite but closed in on itself like a 4-D version of a balloon. The idea is that if you were to travel infinitely in any one direction then you would eventually come back to the point where you started, just as you would if you were confined to travel on the surface of a balloon. Scientific theory therefore seems to at least be consistent with the idea that a spatial dimension cannot be infinite.

For the kalam argument, it is important to note that time, as viewed by the eternalist, is a dimension not unlike the three dimensions of space. Therefore it would also be confined to a finite length, based on the arguments above. Time must have a beginning point, since any finite series has a beginning point. Hence, it is impossible to both believe in the eternalist view of time and to believe in an infinite past.

**Objections**

One objection that may be brought up concerns whether or not time can extend infinitely into the future. If this argument is correct, then wouldn’t it be reasonable to conclude that there must be an end to time? After all, a set with a beginning but no end is just as infinite as a set with an end but no beginning. It seems that this is the conclusion that an eternalist would have to make as well. However, for the presentist, the only objects that are real are those that exist now. Under that view, time may go on into the future but will never actually achieve infinity, and hence it is just a potential infinite. It is important to note that the proofs in this paper with transfinite arithmetic apply only to actual infinities and not potential infinities, since a potential infinite is not infinite at any given moment.

**Summary**

In summary, I have presented the kalam cosmological argument as follows.

1) Right handed subtraction in trans-finite arithmetic leads to realizable contradictions.

2) Any operation that leads to a realized contradiction is metaphysically impossible.

3)Therefore, right handed subtraction of an actual infinite is metaphysically impossible.

4) Right handed subtraction of any realized object is always metaphysically possible.

5) Therefore, actual infinites are not metaphysically possible

6) Time, as viewed by eternalism as an infinite pre-existing dimension, is an actual infinity.

7) Therefore, an infinite dimension of time is metaphysically impossible (from 5-6).

8) Anything that is not infinite is finite.

9) Therefore, time is finite in length (i.e. has a beginning) (from 7-8).

10) Anything that has a beginning has an external cause.

11) Time (and hence the universe) has an external cause (from 9-10).

The eternalist view of time has for some time been used by philosophers as a way around the kalam argument. However, I hold that the arguments presented in this paper show that even if one assumes the eternalist view of time, the kalam argument remains valid support of the existence of God.

**References**

1 Markosian, Ned, "Time", *The Stanford Encyclopedia of Philosophy (Winter 2010 Edition)*, Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/win2010/entries/time/>.

2 Craig et al, *The Blackwell Companion to Natural Theology.* A John Wiley & Sons, pp. 101-201, (2009).

3 A. Grunbaum, *A new cirtique of theological interpretations of physical cosmology*. British Journal for the Philosophy of science, vol 51, pp. 1-43 (2000)

4 Craig et al, *The Blackwell Companion to Natural Theology.* A John Wiley & Sons, pp. 183-184, (2009).

5 J. W. Dauben, *George Cantor: his mathematics and philosophy of the infinite. *Princeton University Press, pp.105-107, (1990)

I realize you said you wrote this when you were just learning about transfinite arithmetic; I'm afraid it shows.

ReplyDeleteI'll address only a couple of points. You write, "If it is true that subtraction with transfinite numbers lead to contradictions, then it is clear that an actual infinite quantity cannot exist for any thing that can be subtracted in the real world." You then proceed from that as though the 'if' has been demonstrated true to construct an argument. Inverse operations are by convention eschewed within Cantor set theory. And we know beyond any doubt that Cantor's set theory had problems to such an extent that it's needed to be repaired a few times over the years; cf ZFC right up through Robinson's standard analysis through to Nelson's internal set theory.

Of particularly curious note is that Ed Nelson is an ultrafinitist, but nevertheless has to accept both infinitesimals and infinite numbers (though he prefers to call infinite numbers simply unlimited) to prove internal set theory. What is more is that these limitless numbers conserve all properties of what he calls 'standard' numbers, and has all the relevant axioms of ZFC. This has been accepted by the mathematical community for better than 30 years now, but tired old religious apologists are still plodding along as though we've not dramatically improved our mathematics since Hilbert and Cantor.

The second point isn't so much a mathematical objection as it is linguistic one. When confronted with real objects that we can group up in various ways, it is the groups which are the operands on which the operations are performed. Consider for a moment a table top with a grouping of 20 pennies. It's only a metaphor to say that we take away x number of pennies when we perform the operation of subtraction on the group of 20. The pennies don't actually go anywhere; they merely are no longer elements of the group from which we've subtracted their number.

The third point is on your 'metaphysics' proposition. Whether one operates on the A or B theory of time, one thing remains true: the present exists and was gotten to through a series of both temporal and spatial events which have already happened. The past is not fundamentally different in nature from the future. The only significant distinction between them is where stand in relation to either - and this persists on whatever notion of time one decides to like. Of course, relativity handily dispensed with the A theory of time.

And if it's incoherent to suppose an infinite set of causal events in the past to get to where we are here today (though this is untrue - there's no problem whatever with a past infinity; simply because we're embedded in some infinity is no argument that we can't arrive at some segment), then so too must it be true for a future infinity.

Even William Lane Craig concedes that the A theory of time is incompatible with contemporary physics (though he quite often makes a big attempt at showing he's right within the mainstream of contemporary physics - but what's a little lying here and there when souls are at stake, eh?). In order to square that particular circle, one has to reset physics to where it was in the 19th century.

Err, Robinson's non-standard analysis that should have read.

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