Defending the Possibility of an Infinite Past

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Defending the Possibility of an Infinite Past

By: Antonio Augusto Senra

Sponsor: Dr. Gene Witmer

April 2014

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Table of Contents

INTRODUCTION ................................................................................................................................ 3

DEFINING “ACTUAL INFINITE” ......................................................................................................... 4

CRAIG’S TWO MAJOR ARGUMENTS ............................................................................................... 11

ARGUMENT FROM ABSURD MATHEMATICAL CONSEQUENCES ................................................... 13

NUMBERED SPINES ARGUMENT ................................................................................................... 16

ARGUMENT FROM INTUITIONISM ................................................................................................. 20

ARGUMENT FROM UNSUPPORTED REVISION OF MATHEMATICS ................................................ 22

TRISTRAM SHANDY ARGUMENT .................................................................................................. 26

COUNTING MAN ARGUMENT ....................................................................................................... 30

FORMATION OF PAST ACTUAL INFINITE ARGUMENT .................................................................. 33

ARGUMENT FROM ORDINAL NUMBERS ....................................................................................... 35

INFINITE DIVISIBILITY OF TIME ARGUMENT ............................................................................... 42

CONCLUSION ................................................................................................................................. 46

BIBLIOGRAPHY ............................................................................................................................. 47

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INTRODUCTION

Events in time either had a beginning, or they occur infinitely far back into the past.

These are the only possibilities for the nature of events in time, and only one of them can be

correct. To affirm that events in time occur infinitely far back into the past is to deny that there

was a first event. Conversely, to deny that events in time occur infinitely far back into the past is

to affirm that there was a first event. William Lane Craig advocates for the position that time had

a beginning, and his strategy is to give multiple arguments endeavoring to prove that events in

time do not occur infinitely far back into the past. In this paper, I will reject Craig’s arguments

and conclude that it is possible that events in time occur infinitely far back into the past.

Before beginning, however, I will draw attention to the significance of this issue. The

Temporal Cosmological Argument is one of the strongest arguments in support of theism1. In its

most simplistic representation, the Temporal Cosmological Argument claims the following:

If there was a cause of the first event in time, God must have been responsible for that

first cause. There must have been a cause of the first event in time, so God must be

responsible for that cause.

1 The Cosmological Argument exists in two popular versions: the Temporal Cosmological Argument and the

Contingency Cosmological Argument. The Temporal Cosmological Argument is dependent upon the nature of time

while the Contingency Cosmological Argument is not. Since this paper is concerned with the nature of time, it will

have tremendous significance for the Temporal Cosmological Argument and much less so for the Contingency

Cosmological Argument. For further reading on the Temporal Cosmological Argument, one may consult William

Lane Craig’s book, The Kalam Cosmological Argument (see Bibliography for full citation). For further reading on

the Contingency Cosmological Argument, one may consult Chapter 11 of Metaphysics by Richard Taylor (see

Bibliography for full citation).

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This argument hinges on the Premise that there must have been a first event, and that this event

had a cause. If events in time occur infinitely far back into the past, however, then there would

have been no first cause of events in time. So, in order for the theist to appeal to the Temporal

Cosmological Argument, they must have reason for believing that events in time had a

beginning. The greater the likelihood that events in time occur infinitely far back into the past,

the weaker the force of the Temporal Cosmological Argument. This is why Craig offers multiple

arguments against the claim that events in time occur infinitely far back into the past. However,

if I am successful in showing that those arguments do not work, I strengthen the likelihood that

events in time occur infinitely far back into the past. The more I strengthen this likelihood, the

weaker the Temporal Cosmological Argument becomes; and, by extension, the weaker the

argument for theism becomes.

DEFINING “ACTUAL INFINITE”

Our intuitive notion of “infinity” is to think of a collection of things that has no limit.

However, to properly understand infinity, we must go beyond our intuitive notion of infinity, and

become clear on its technical meaning. In 1877, George Cantor crafted a precise definition of

infinity with his work in set theory. Understanding this definition is fundamental to considering

many arguments against infinity. For this reason, we must first become familiar with key

terminology.

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Cardinal Number: The measure of the size of a set (i.e., the number of elements in a set).

For example, the set {1, 2, 3} has a cardinal number of three. The set {20, 34, 59, 88} has

a cardinal number of four. The set {5, 19, 21, 44, 51} has a cardinal number of five.

One-to-one correspondence: A property given to a set, set A¸ when there is a way of

associating elements in set A with elements in another set, set B, such that every element

in set A is associated with one and only one element in set B, and every element in set B

is associated with one and only one element in set A. If two sets have a one-to-one

correspondence, they have the same cardinal number. Here is an example of two sets with

a one-to one correspondence:

Equivalent: Two sets are equivalent if and only if they have the same cardinal number.

(Moreover, sets having a one-to-one correspondence are equivalent, since sets having a

one-to-one correspondence have the same cardinal number.)2

2 “Equivalent” can sometimes have other meanings; however, any time I use “equivalent” in this paper, I am

referring to this definition.

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At this point we know enough to talk about certain properties of finite sets. Imagine we have Set

A: {1, 2, 3}. We can count the number of elements and discover that this set has a cardinal

number of three. Imagine another set, Set B: {11, 12, 13}. Set B can be placed in a one-to-one

correspondence with Set A. This means that the two sets are equivalent. Since we know Set A

has a cardinal number of three and Set B is equivalent to Set A, then we know Set B has a

cardinal number of three. Now let us introduce another term.

Aleph-null (N0): The cardinal number of a set that is equivalent to the set of natural

numbers.3

Unlike finite sets, infinite sets cannot be represented by a finite number. Aleph-null is the

“number”4 used to represent the cardinality5 of an actually infinite set.

When we discussed finite sets above, we began by counting the elements in Set A, and

we determined that it had a cardinal number of three. However, we cannot “count the elements”

in an infinite set. For this reason, our definition starts by acknowledging that the set of natural

numbers is an actually infinite set. There is no proof for this claim; rather, it is the starting point

of our reasoning. From this claim we can deduce the infinitude of other sets. If we determine that

a set, Set C, can be placed in one-to-one correspondence with the set of natural numbers, then we

3All infinite sets have a cardinal number of at least aleph-null, but there are some that have an even greater cardinal

number. Those infinite sets are not the ones this paper is concerned with. The distinction between those infinite sets

and the ones of this paper’s concern lies in the distinction between a countably infinite set and an uncountably

infinite set. For further reading on the distinction between these types of infinite sets, one may consult Craig’s book,

The Kalam Cosmological Argument, pages 70-82 (see Bibliography for full citation). 4 The reason I use scare quotes around “number” is because aleph-null does not have all the same properties of a

number. To think of it as acting in the same way as “9,” “27,” or “34” can be problematic. It is important to

remember that aleph-null is not a “number” in the same way as what are commonly considered numbers. 5 “Cardinality” is another way of saying “cardinal number.”

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have determined that Set C is equivalent with the set of natural numbers. Since the set of natural

numbers has a cardinal number of aleph-null, set C has a cardinal number of aleph-null.

Now we can introduce the final missing piece which will allow us to apply a clear,

succinct definition of an actually infinite set:

Proper Subset: A subset of a set which does not have all of the elements contained in the

original set. Here are some illustrations:

{1} is a Proper Subset of {1, 2, 3, 4, 5}

{3, 4, 5} is a Proper Subset of {1, 2, 3, 4, 5}

{1, 2, 3, 4, 5} is a Subset, but not a Proper Subset of {1, 2, 3, 4, 5}

With this term in mind, here is the definition of an actually infinite set.

Actually Infinite Set: A set is actually infinite if and only if it has a proper subset which is

equivalent to it.

Here is an example of a set that is actually infinite:

A = {1, 2, 3, 4, 5, 6, 7, …}

We can see that set A is actually infinite by recognizing that it has a proper subset which is

equivalent to it:

B = {3, 4, 5, 6, 7, …}

Their equivalence is evident by the fact that Set B can be placed in one-to-one correspondence

with Set A. We could associate 1 from Set A with 3 from Set B, 2 from Set A with 4 from Set B,

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and so on. Each member would paired with exactly one member of the other set, and no member

would be left out. Since Set A has a proper subset that is equivalent to it, Set A is an actually

infinite set. Furthermore, Set B is an actually infinite set as well. This can be seen when we take

the following proper subset of Set B:

C = {5, 6, 7, …}

Their equivalence is evident by the fact that Set C can be placed in one-to-one correspondence

with Set B. Since Set B has a proper subset that is equivalent to it, Set B is an actually infinite

set.

Here is an example of a set that is not actually infinite:

{1, 2, 3, 4, 5}

This set has a cardinality of five. There is no proper subset of this set which also has a cardinality

of five.

As you may have noticed, in this paper I have been referring to an infinite set as “an

actually infinite set,” rather than simply “an infinite set.” This is to avoid the distinction that

Craig outlines between potential infinite and actual infinite, which I will now elaborate.

All potentially infinite sets have a cardinality which is described with a natural number.

An actually infinite set has a cardinality of at least aleph-null. Aleph-null is not a natural number.

So, if a set has a cardinality represented by a natural number, then it cannot have a cardinality of

aleph-null, and thus, it cannot be an infinite set. The reason we refer to it as “potentially infinite”

is because there seems to be nothing which precludes it from continuing to increase; however, no

matter how much it increases, at any given time we can look at a potentially infinite set and

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determine the finite number which corresponds to its cardinality To better understand a potential

infinite, imagine a man who, at this very moment, begins sequentially counting the natural

numbers. The man could continue counting endlessly, but at any given point we could stop him

and determine the amount of numbers he has counted. In this example, the last number the man

counted also represents the total amount of numbers that the man has counted. Since we could

always assign a finite number to the amount of numbers that the man has counted, the man

would never have a set with a cardinality of aleph-null. Consequently, this man would never

have completed reciting an actually infinite collection of numbers.

An actually infinite set, on the other hand, does not have a cardinality which can be

described with a finite number. Its cardinality instead is at least aleph-null. To continue with the

example of the counting man, instead of a man who begins counting now and continues

endlessly, the relevant example here is of a man who has been counting backwards from eternity

(this could be represented like this: {…, -8, -7, -6, …} ). Regardless of which number this man is

on, you could not say how many numbers he has counted. The key difference to remember is that

an actually infinite set truly has a cardinality of at least aleph-null, while a potentially infinite set

has a cardinality that can be determined with a natural number. In this paper, I will be concerned

with an actually infinite set, not a potentially infinite set.6

Now let us consider an actual infinite in time. There are two ways which we can make

this consideration. The first way is to place ourselves at the present and extend time into the

future.

6 Clearly all actually infinite sets are also potentially infinite sets. When I say that I am not concerned with

potentially infinite sets, I mean sets that are merely potentially infinite sets, (i.e., sets that are not actually infinite).

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The second way is to place ourselves at the present and extend time into the past.

The primary concern of this paper will be with an actual infinite in time that extends

limitlessly far back into the past. For ease in exposition, I will henceforth use the following

abbreviation:

PAI (Past Actual Infinite): An actual infinite in time that extends limitlessly far back into

the past. To facilitate conceptualization, one can imagine the PAI as a set that has the

following form: { …, -3, -2, -1, 0}, with “0” representing the current moment in time.

Also, it is important to note that there is some finite temporal distance between any two

moments of this set.

Some of Craig’s arguments will be against an actual infinite in general. Others will be

specifically against the PAI.

One final preliminary matter: in three of Craig’s arguments (Argument from Intuitionism,

Argument from Unsupported Revision of Mathematics, and Argument from Ordinal Numbers),

he concludes the impossibility of an actual infinite without qualification. That is, he claims that

an actual infinite is impossible even in mathematics. For the remainder of his arguments, he

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relies upon the distinction between an actual infinite existing in the mathematical realm and an

actual infinite existing in reality. He describes this distinction as follows:

What I shall argue is that while the actual infinite may be a fruitful and consistent concept

in the mathematical realm, it cannot be translated from the mathematical world into the

real world, for this would involve counterintuitive absurdities (Craig, 1979, 69)

The object of my paper will be to say that Craig is wrong. I will argue it is possible both that an

actual infinite exists in the mathematical realm and that an actual infinite exists in reality. More

specifically, I will say that it is possible that a particular kind of actual infinite, the PAI, exists in

reality.

CRAIG’S TWO MAJOR ARGUMENTS

William Lane Craig gives two “major” arguments for the impossibility of the PAI. The

reason I call them “major” arguments is because these two arguments arrive at his ultimate

conclusion: that the PAI is impossible. All his arguments besides these two merely serve to

strengthen premises of these major arguments. As you will soon see, both of the major

arguments clearly entail the impossibility of the PAI. Craig’s other arguments do not entail the

impossibility of the PAI as clearly. With those arguments, it will be necessary to apply their

conclusions to one of the major arguments to elucidate their role in proving the impossibility of

the PAI.

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Fortunately, Craig lays out both major arguments formally. Both of these formal

statements are taken directly from Craig’s work. He refers to them repeatedly throughout his

book, The Kalam Cosmological Argument (see Bibliography for full citation).

First Major Argument:

1. An actual infinite cannot exist. [Premise]

2. A beginningless series of events in time is an actual infinite. [Premise]

C. Therefore, a beginningless series of events in time cannot exist. [1, 2]

This argument is straightforward. There is no questioning its validity. It is also easy to see that if

its conclusion is true, then the PAI cannot exist.

The Second Major Argument has a different conclusion, but is still as damning for the

possibility of the PAI as the First Major Argument.

Second Major Argument:

1. The series of events in time is a collection formed by adding one member after another.

[Premise]

2. A collection formed by adding one member after another cannot be actually infinite.

[Premise]

C. Therefore, the series of events in time cannot be actually infinite. [1, 2]

This is as straightforward as the First Major Argument, so once again, there is no room to

question its validity. Its conclusion also clearly entails the impossibility of the PAI.

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Armed with these two major arguments, we can now broach Craig’s other arguments.

These other arguments are supplementary to the major arguments. In effect, Craig relies on the

pairing of the major arguments and these supplementary arguments to achieve his goal of

exposing the impossibility of the PAI. The two major arguments have some premises which are

not obviously correct; so, by themselves, we would not be convinced of their conclusions. Craig

relies on the supplementary arguments give us reason to believe those premises, and ultimately,

their conclusions. On the other hand, the supplementary arguments by themselves are insufficient

to arrive at Craig’s ultimate conclusion that the PAI is impossible; so Craig relies on the major

arguments to reach his ultimate conclusion. Since the supplementary arguments and the major

arguments work in unison, as I progress through each of Craig’s supplementary arguments, I will

explain their place within the context of the major arguments.

ARGUMENT FROM ABSURD MATHEMATICAL CONSEQUENCES

This argument functions as support for the first Premise of the First Major Argument:


This argument takes the form of a Reductio Ad Absurdum argument. Craig will attempt to show

that if an actual infinite is existed, then it would necessarily entail a situation which is both

possible and impossible (which is a logical contradiction). If a situation necessarily entails a

logical contradiction, it must be impossible. Therefore, if Craig shows that an actual infinite

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necessarily entails a situation which is both possible and impossible then he will have succeeded

in showing that an actual infinite is impossible.

Craig describes the absurd mathematical consequence as follows:

… What is infinity minus infinity? Well mathematically, you get self-contradictory

answers. For example, if you subtract all the odd numbers {1, 3, 5, …} from all the

natural numbers {0, 1, 2, 3, …}, how many numbers do you have left? An infinite

number. So infinity minus infinity is infinity. But suppose instead you subtract all the

numbers greater than 2—how many are left? Three. So infinity minus infinity is 3! It

needs to be understood that in both these cases we have subtracted identical quantities

from identical quantities and come up with contradictory answers. (Craig and Armstrong,

2004, 4)

Craig argues that because subtraction of an actual infinite from another actual infinite can yield

different results, an actual infinite cannot be possible. To visualize this apparent problem with

subtracting actually infinite sets, here is a diagram (all the sets in this diagram are actually

infinite, and thus, have a cardinality of aleph-null):

{1, 2, 3, …} – {1, 2, 3, …} = 0

{1, 2, 3, …} – (2, 3, 4, …} = 1

{1, 2, 3, …} – {3, 4, 5, …} = 2

{1, 2, 3, … } – {2, 4, 6, …} = aleph-null

(Craig, 1979, 81)

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Here is my formal reconstruction of Craig’s argument:

1. If actually infinite sets were possible, it would be possible to subtract a set with a

cardinality of aleph-null from another set with a cardinality of aleph null and yield a set

whose cardinality is 0. [Premise]

2. If actually infinite sets were possible, it would be possible to subtract a set with a

cardinality of aleph-null from another set with a cardinality of aleph null and yield a set

whose cardinality is 1. [Premise]

3. If actually infinite sets were possible, it would be possible for the cardinality of a set that

results from the subtraction of one set from another to depend on something more than

the cardinality of the two sets involved in the subtraction. [1, 2]

4. It is impossible for the cardinality of a set that results from the subtraction of one set

from another to depend on something more than the cardinality of the two sets involved

in the subtraction. [Premise]

5. If actually infinite sets were possible, it would be both possible and impossible for the

cardinality of a set that results from the subtraction of one set from another to depend on

something more than the cardinality of the two sets involved in the subtraction. (Logical

Contradiction) [3, 4]

6. A situation cannot be possible if it entails a logical contradiction. [Premise]

C. Actually infinite sets are impossible. [5, 6]

The problem in Craig’s argument is Premise four. This Premise may seem intuitive, but we must

remember that actually infinite sets do not behave identically to finite sets. Furthermore, the

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difference between actually infinite sets and finite sets is not incidental. Actually infinite sets

were integrated into set theory with the understanding that it has fundamental differences from

finite sets. To believe that actually infinite sets must conform to rules that govern finite sets is to

crucially misunderstand set theory. Once we recognize that actually infinite sets are not governed

by all of the same rules as finite sets, we can accept that Premise four is incorrect.

Another important point to remember is that most mathematicians have embraced set

theory; so, by extension, most mathematicians have embraced the nature of actually infinite sets.

These mathematicians have not objected to the results of subtraction between two actually

infinite sets. This should allow us to believe that certain rules that apply to finite sets do not

apply to actually infinite sets. As a result, we have further reason to deny Premise four, and

ultimately, the argument’s conclusion that “actually infinite sets are impossible.”

NUMBERED SPINES ARGUMENT



This argument takes the form of a Reductio Ad Absurdum argument just like the Argument from

Absurd Mathematical Consequences. Craig will attempt to show that if an actual infinite were

possible, it would necessarily entail a situation which is both possible and impossible (which is a

logical contradiction). In this particular argument, the “situation” in question is a library with an

infinite collection of books. If this library example entails a logical contradiction, it must be

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impossible. Therefore, if Craig shows that this library example necessarily entails a situation

which is both possible and impossible, he will have succeeded in showing that it is impossible.

Since the possibility of an actual infinite requires that the library example is also possible, if

Craig shows that the library example is impossible, he will have shown that an actual infinite is

impossible.

Craig asks us to imagine a library with an infinite number of books. On the spine of each

book there is a natural number. Since there are an infinite number of books and an infinite

amount of natural numbers, there is a one-to-one correspondence between the set of books and

the set of natural numbers. This means that for every natural number, there exists a book in that

library that corresponds to it. For Craig, the absurdity in this situation is that you could not add a

book to that library with a new natural number printed on its spine. Here is a condensed version

of the argument in his own words:

Suppose … that each book in [an infinite] library has a number printed on its spine so as

to create a one-to-one correspondence with the natural numbers. Because the collection is

actually infinite, this means that every possible natural number is printed on some book.

Therefore, it would be impossible to add another book to this library. For what would be

the number of the new book? Clearly there is no number available to assign to it. Every

possible number already has a counterpart in reality, for corresponding to every natural

number is an already existent book. Therefore, there would be no number for the new

book. But this is absurd, since entities that exist in reality can be numbered. (Craig, 1979,

83)

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This argument represents a very common type of argument against a PAI. It reveals a strange

circumstance that could be possible if an actual infinite were possible; but characterizes the

circumstance as an absurdity. It is important to note that to construct an effective Reductio Ad

Absurdum argument, one must expose two logically contradictory statements that follow from

the initial condition under consideration (in this case, the condition under consideration is the

existence of a library with infinitely many books, each with a unique natural number printed on

its spine). To merely point to a situation which seems counterintuitive is not sufficient.

I will try to reconstruct Craig’s argument in a formal presentation. In this reconstruction,

I attempt to highlight the most likely contradiction that Craig is after. When we closely scrutinize

the formal presentation of Craig’s argument, we will see that there is no logical contradiction that

necessarily follows from the initial condition under consideration (Premise 1)

1. Imagine a library with an infinite number of books and each of the books has a unique

natural number printed on its spine. [Premise]

2. The set of natural numbers is an infinite set. [Premise]

3. If two sets are infinite, there exists a one-to-one correspondence between them. [Premise]

4. If the library example were possible, there would exist a one-to-one correspondence

between the set of books in the library and the set of natural numbers. [1, 2, 3]

5. If there exists a one-to-one correspondence between the set of books in the library and

the set of natural numbers, it would be impossible to add a book to the library that

possesses a unique natural number on its spine. [Premise]

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6. If the library example were possible, it would be impossible to add a book to the library

that possesses a unique natural number on its spine. [4, 5]

7. If the library example were possible, it would always be possible to add a book to the

library that possesses a unique natural number on its spine. [Premise]

8. If the library example were possible, it would be both possible and impossible to add a

book to the library that possess a unique natural number on its spine. (Logical

Contradiction) [6, 7]


C. The library example is impossible. [8, 9]

Upon first glance it would seem that Premise 6 and Premise 7 represents the logical contradiction

that necessarily follow from the initial situation (that the PAI is possible). The problem in this

argument arises because we do not have sufficient reason to believe Premise 7 is true. It may

simply be the case that it is not always possible to add a book to the library that possesses a

unique natural number on its spine. There is no axiom anywhere in logic or mathematics that

says that Premise 7 is true. In fact, with an intimate understanding of an actual infinite, we can

see that it is perfectly conceivable to deny this Premise. Our hypothetical library has a limitless

collection of books. Therefore, the books would correspond with the limitless natural numbers. It

makes complete sense to believe that one could not add a new natural number. Perhaps the

“strange-ness” can be alleviated even further if we remember that this does not mean a person

could not add a new book altogether. It simply means that they could not add a book that had a

unique natural number on it. Therefore, since Premise 7 is not necessarily true, we do not have

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reason to believe that a logical contradiction necessarily follows from the PAI being possible. So,

we do not have reason to believe Craig’s Reductio Ad Absurdum argument.

ARGUMENT FROM INTUITIONISM



Intuitionism is a school of mathematics that holds that true mathematical statements must

be derived as a result of the constructive mental activity of humans and not through fundamental

principles claimed to exist in an objective reality (See Wikipedia, 2014a). Intuitionists deny any

purported mathematical claims which cannot be derived as a result of the constructive mental

activity of humans. Intuitionists argue that an “actual infinite” cannot be derived as the result of

the constructive mental activity of humans; therefore, an actual infinite cannot be said to exist.

Borrowing the Intuitionist view, Craig says:

We have seen that the root presupposition of intuitionism is that the basis of mathematics

is found in the pure intuition of counting. This, constructability by actual operations

becomes the prerequisite of any legitimate mathematical operation. Since an actual

infinite cannot be constructed by the human mind, it follows that the infinite is not a well-

defined totality. And not being well defined, the actual infinite cannot be said to exist in

the mathematical realm (Craig and Smith, 1993, 21-22).

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Here is my formal reconstruction of this argument:

1. An adequate definition of a mathematical claim must be derived as a result of the

constructive mental activity of humans. (The Intuitionist view) [Premise]

2. No constructive mental activity of humans can be used to define an actual infinite.

[Premise]

3. Hence, "actual infinite" cannot be adequately defined. [2, 3]

4. If a mathematical term cannot be adequately defined, we cannot assume its existence in

mathematics. [Premise]

C. Hence, we cannot assume the existence of an actual infinite in mathematics. [1, 4]

The first Premise is the weakest point of this argument. However, attacking it amounts to

attacking Intuitionism as a school of mathematics. This would require a considerable amount of

argumentation. As a result, I will not offer a comprehensive attack of Intuitionism. However, this

argument is still among the strongest arguments which Craig offers, so I will give two responses

which provide reasons for one to doubt Intuitionism.

Perhaps the strongest point against Intuitionism is that it is the view of a small minority

of mathematicians. The lack of support for Intuitionism by those professionally studying

mathematics suggests there are strong reasons to doubt it. Furthermore, the most commonly

accepted school of mathematics is the Zermelo–Fraenkel set theory; acceptance of Intuitionism

would entail denying many of this school’s principles (See Wikipedia, 2014a).

Another reason to doubt Intuitionism is that it seems particularly weak when offering an

explanation of infinity. The spirit of Intuitionism is to ensure that all mathematical statements are

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reducible to ideas we can conceive of intuitively. In denying the possibility of infinity, the

Intuitionist is claiming that infinity is not something we can conceive of intuitively. This does

not seem to be true. We can intuitively conceive of what it would mean for a collection to be

actually infinite. As we stated before, it would be a collection of things that has no limit. So the

Intuitionist view seems to contradict its underlying intentions when it declares that an actual

infinite is not possible. This type of contradiction contributes to the doubt that we ought to have

towards its claims.

In responding to the Argument from Intuitionism, the debate shifts to whether or not one

should believe in Intuitionism. Admittedly, my response is not a thorough analysis of this debate.

However, it is important to remember that before embracing this argument as a strong argument

against the PAI, one must be willing to claim the Intuitionist school of mathematics as the school

they find most convincing, and this is a price many will be unwilling to pay.

ARGUMENT FROM UNSUPPORTED REVISION OF MATHEMATICS



Before set theory appeared, there were several mathematical principles which appeared to be

indubitable. Among these were “Euclid’s common notions,” five principles that the

mathematician, Euclid, laid out in his famous mathematical treatise, Elements. The fifth of these

common notions is of great interest to the topic of infinity.

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The whole is greater than the part (Wikipedia, 2014b)

When Cantor emerged with set theory, Euclid’s fifth notion proved to be contradictory to set

theory. If the fifth notion were true, then set theory could not define an actually infinite set.

Recall the definition of an actually infinite set:

A set is actually infinite if and only if it has a proper subset which is equivalent to it.

This means that an actually infinite set will have a whole that is equal to the part, which stands in

direct contradiction to Euclid’s fifth notion. Cantor’s solution was to deny that Euclid’s fifth

common notion applied to infinite sets. Instead, he appeals to the following principle:

If two sets can be placed in a one-to-one correspondence, then those two sets are equivalent.

This principle is, of course, the foundation for our current definition of infinity.

Craig objects to Cantor’s move. He argues that the reasons behind the claim that

“Euclid’s fifth common notion does not apply to infinite sets” is wholly unfounded. Craig says

that one can just as easily make the claim that “the Principle of Correspondence7 does not apply

to infinite sets.” In essence, Craig is saying that there is no reason to reject Euclid’s fifth

common notion over rejecting the Principle of Correspondence. Furthermore, if Cantor had

rejected the Principle of Correspondence rather than the Euclid’s fifth common notion, then an

actual infinite would be impossible. This is precisely the consequence that Craig desires, so it is

no surprise that he seeks justification for Cantor’s choice. Here is an excerpt where Craig

explains this dilemma in his own words:

7 “Principle of Correspondence” is the term Craig uses to refer to the principle Cantor uses in his definition of an

infinite set: if two sets can be placed in a one-to-one correspondence, then those two sets are equivalent.

24

But which principle is to be sacrificed? Both seem to be intuitively obvious principles in

themselves, and both result in counter-intuitive situations when either is applied to the

actual infinite. The most reasonable approach to the matter seems to be to regard both

principles as valid in reality and the existence of an actual infinite as impossible. (Craig

and Smith, 1993, 24)


1. In order to claim that an actual infinite is possible, we must have reason to believe that

either Euclid's maxim does not apply to it or the Principle of Correspondence does not

apply to it. [Premise]

2. We have no reason to believe that Euclid's maxim does not apply to an actual infinite.

[Premise]

3. We have no reason to believe that the Principle of Correspondence does not apply to an

actual infinite. [Premise]

C. We cannot claim that an actual infinite is possible. [1, 2, 3]

I will now argue against the second Premise to this argument. Let us begin by considering the

purpose of these two principles (Euclid’s fifth common notion and the Principle of

Correspondence). As Craig says, these two are supposed to be “intuitively obvious principles in

themselves.” There are no proofs which establish their existence a priori. We consider them to

be among the starting points in mathematics. A similar claim, for example, is: “a triangle is a

plane figure with three sides.” There is no proof for this statement; it is simply accepted as a

starting principle.

25

With the understanding that neither of these statements will have a conclusive proof

which establishes their applicability, I can now proceed into arguments for why Euclid’s fifth

common notion ought to be discarded in favor of the Principle of Correspondence.

I will begin with an appeal to intuition. Consider again the intuitive understanding of

infinity: a collection that has no limit. When we are presented with two collections that have no

limit, we would never say that one collection was greater than the other. Instead, we would refer

to both collections as infinite. The consequence of this is that a collection that “has no limit” will

have portions of itself which are “as big” as it. This intuitive understanding eschews the idea that

“the whole is greater than the part.” From this we see that we do have reason to believe that

Euclid's fifth common notion does not apply to an actual infinite (this stands in direct opposition

to Premise 2).

Another argument against Euclid’s fifth common notion can be made by appealing to the

current strength of set theory. Current set theory dictates that in the consideration of infinite sets,

we should reject Euclid’s fifth common notion in favor of the Principle of Correspondence. This

practice has, thus far, allowed set theory to flourish as a successful foundation system for

mathematics. By contrast, no mathematician has been able to craft a set theory in a way that

dictates that in the consideration of infinite sets, we should reject the Principle of

Correspondence in favor of Euclid’s fifth common notion. In other words, the Principle of

Correspondence has been proven to be successful in allowing set theory to thrive, while Euclid’s

fifth common notion has not. This certainly gives us a reason to believe that Euclid's fifth

common notion does not apply to an actual infinite (this stands in direct opposition to Premise

2).

26

These two arguments in conjunction give us a meaningful amount of evidence in favor of

the belief that Euclid’s fifth common notion does not apply to an actually infinite set. Keep in

mind that my response does not suggest that we have reason to reject Euclid’s fifth common

notion altogether, but simply that we have reason to reject that Euclid’s fifth common notion

applies to an actually infinite set.

TRISTRAM SHANDY ARGUMENT



This argument has a similar structure to the Numbered Spines Argument. It argues that if the PAI

were possible, it would necessarily entail a situation which is both possible and impossible

(which is a logical contradiction). The situation in question in this argument is what Craig calls

the “Tristram Shandy example” (Craig, 1979, 97). If Craig is able to show that the Tristram

Shandy example results in a logical contradiction, it must be impossible. Craig argues that since

the PAI requires that the Tristram Shandy example be possible, if the Tristram Shandy example

is impossible, then the PAI is also impossible.

In this hypothetical, Tristram Shandy is the name of a man who has existed from eternity.

That is to say, just like time, Tristram Shandy never began existing; he has always existed. The

important detail about Tristram Shandy is that he is writing an autobiography about every day of

his life; however, it takes him one full year to finish writing about one day. Furthermore, he only

27

starts writing about the next day of his life after he finishes writing about the day which came

before it. To see this a different way: at the end of his n-th year of writing, Tristram Shandy will

have finished writing about day n. So, at the end of year 1, Tristram Shandy will have finished

writing about day 1; and, at the end of year 100, Shandy will have finished writing about day

100. However, as Tristram Shandy writes about day n, there are several days that elapse. This

creates an apparent circumstance where Tristram Shandy is perpetually falling behind in his

work. At the end of year 2, Shandy has finished writing about day 2; but by this point, 730 days

have passed. This leads Craig to ask the question:

Suppose Tristram Shandy has been writing from eternity past at the rate of one day per

year. Would he now be penning his final page? (Craig, 1979, 98)

I argue that the answer to Craig’s question is “no.” However, Craig identifies what, at first,

seems to be a strange circumstance that necessarily follows from the Tristram Shandy

hypothetical. The Principle of Correspondence commits us to the claim that the number of days

Tristram Shandy has lived is equivalent to the number of days Tristram Shandy has written

about. After all, both of these sets (number of days lived and number of days written about) are

infinite sets with a cardinality of aleph-null. As a result, it would seem as though, if we were to

stop Shandy’s writing at this very moment, he would have written about the final day of his life.

This seems problematic given that we know that Tristram Shandy requires an entire year to write

about a single day. To say nothing of the fact that, given that Shandy falls perpetually behind in

his work, he has many other days to write about before he can even think of writing about today.

Craig says this is absurd. The question “would he now be penning his final page?” express his

suspicion about this entire hypothetical. He argues:

28

According to the [Principle of Correspondence] … a one-to-one correspondence between

days and years could be established so that given an actual number of years, the book will

be completed. But such a conclusion is clearly ridiculous, for Tristram Shandy could not

yet have written today’s events down. In reality, he could never finish, for every day of

writing generates another year of work. But if the principle of correspondence were

descriptive of the real world, he should have finished—which is impossible. (Craig, 1979,

98)

Here is my formal reconstruction of Craig’s argument.

1. The Tristram Shandy example describes a PAI where an individual exists from eternity

up through today and he has been writing an autobiography about his life at a pace of

“one day written about” per year. [Premise]

2. If the Tristram Shandy example is possible, today is a day lived but not a day written

about. [1]

3. If the Tristram Shandy example is possible, there is a day in the set of days lived that is

not in the set of days written about. [2]

4. If there is a day in the set of days lived that is not in the set of days written about, then the

sets are not equivalent. [Premise]

5. If the Tristram Shandy example is possible, the set of days lived and the set of days

written about are not equivalent [3, 4]

6. If the Tristram Shandy example is possible, then both the number of days written about

and the number of days lived constitute actually infinite sets. [1]

29

7. Any two sets that are actually infinite are equivalent. [Premise]

8. If the Tristram Shandy example is possible, then the set of days lived and the set of days

written about are equivalent. [6, 7]

9. If the Tristram Shandy example is possible, then the set of days lived and the set of days

written about is both equivalent and not equivalent. (Logical Contradiction) [4, 8]

10. A situation cannot be possible if it entails a logical contradiction.

C. The Tristam Shandy example is impossible. [5, 10]

I disagree with the fourth Premise of Craig’s argument (if there is a day in the set of days lived

that is not in the set of days written about, then the sets are not equivalent). This Premise

mischaracterizes the definition of “equivalent.” Let us reconsider the Tristram Shandy example

for a moment. If we were to stop him today, then there is certainly no way that he could write

about today in his autobiography. This does not entail that the set of “days lived” and the set of

“days written about” are non-equivalent. Craig is attempting to pair a particular day (today) with

the page in Tristam Shandy’s autobiography that would be about this particular day. One-to-one

correspondence does not demand that a specific element of one set aligns with a specific element

of another set. Rather, it simply demands that all elements of one set align with some element of

another set (regardless of which element it is). Imagine that January 1st was the last day that

Tristram Shandy lived. The fact that January 1st cannot be placed in direct correspondence with

the narrative about January 1st in Tristram Shandy’s autobiography, does not mean that it cannot

be placed in direct correspondence with any day in Tristram Shandy’s autobiography. To

believe, as Craig does, that January 1st must correspond to the narrative that would follow

January 1st is to confuse the nature of one-to-one correspondence. The only relevant factor in

30

determining that two sets are equivalent is whether or not the two sets have the same cardinal

number. In this case, because the two sets are actually infinite, they both have a cardinal number

of aleph-null; therefore, their elements can be placed in a one-to-one correspondence and they

are equivalent. This conclusion directly opposes the fourth Premise, and thus, we have shown

Craig’s argument to be unsound.

COUNTING MAN ARGUMENT



This argument aims for the same effect as the Tristram Shandy Argument and the Numbered

Spines Argument. It invokes a hypothetical example that must be possible if the PAI is possible.

In other words, if it is possible for the PAI to exist, then it must be the case that the following

hypothetical is possible. Craig will aim to show that this hypothetical is impossible. If he

succeeds, then he will have succeeded in showing that the PAI is impossible.

Here is Craig’s explanation of this hypothetical:

[S]uppose we meet a man who claims to have been counting [backwards] from eternity,

and now he is finishing: -5, -4, -3, -2, -1, 0. Now this is impossible. For, we may ask, why

didn’t he finish counting yesterday or the day before or the year before? By then an

infinity of time had already elapsed, so that he should have finished. The fact is, we could

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never find anyone completing such a task because at any previous point he would have

already finished. But what this means is that there could never be a point in the past at

which he finished counting. In fact, we could never find him counting at all. For he

would have already finished. But if no matter how far back in time we go, we never find

him counting, then it cannot be true that he has been counting from eternity. This shows

once more that the series of past events cannot be beginningless. (Craig, 1980)

As I explained in the Numbered Spines Argument section, in order to offer a successful Reductio

Ad Absurdum argument, there must be two contradictory statements that necessarily follow from

the argument’s initial condition. In this excerpt from Craig’s work, there is no straightforward

contradiction. I will now lay out the formal statement of an argument which I believe captures

the contradiction that Craig intends:

1. If the counting man example were possible, there would be some explanation of why he

reached 0 at noon today instead of some earlier time. [Premise]

2. If the counting man example were possible, there could not be an explanation of why he

reached 0 at noon today instead of some earlier time. [Premise]

3. If the counting man example were possible, there both would be and could not be an

explanation of why he reached 0 at noon today instead of some earlier time. (Logical

Contradiction [1, 2]


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C. The counting man example is impossible. [3, 4]8

The problem with this argument is the first Premise. Craig’s justification for this Premise seems

to be an appeal to the Principle of Sufficient Reason9. However, Craig distorts the demands of

the principle. The principle does not demand that we explain why one thing happens instead of

another, it simply demands that we explain why one thing happens. So, we do not need an

explanation for why he reached 0 at noon today instead of some earlier time. We simply need an

explanation for why he reached 0 at noon today. Such an explanation is not complex: the man

“reached 0 at noon today” because “he was at -1 at the moment before noon today when he

reached 0, and he is counting backwards.” If one were to seek an explanation for why he

“reached -1 at the moment before noon today,” then one could simply respond that “he was at -2

at the moment before he reached -1, and he is counting backwards.” One could offer this same

explanation for any number in the sequence of numbers that the counting man counts. What

gives Craig’s argument the appearance of strength is the initial belief that we must have some

explanation for why he reached 0 at noon today instead of some earlier time; but once we realize

that this explanation is not required, Craig’s argument loses its force.

8 This was taken from Dr. Gene Witmer’s “Philosophy of Religion” class of 2013. Thanks to him for providing this

representation. 9 A philosophical principle which states: For every fact F, there must be an explanation why F is the case. (Stanford

University, 2010)

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FORMATION OF PAST ACTUAL INFINITE ARGUMENT

This argument functions as support for the second Premise of the Second Major Argument:


[Premise]

In this argument, Craig analyzes the possible ways that the PAI could have formed and realizes

that none are possible. From this, he concludes that the PAI cannot exist.

Craig divides the formation of the PAI into two categories: as either from successive

addition, or not from successive addition. By successive addition, he means:

[The PAI] is a collection that is instantiated sequentially or successively in time, one

event following upon the heels of another. (Craig, 1979, 103)

Craig argues that to say that the PAI is not formed by successive addition is to say that it is

always infinite. Thus, Craig says that the PAI can either be: formed by successive addition or

formed in a way such that it was always infinite. Then he claims that the latter is impossible:

If the series or collection of events is always infinite … it would be just given at some

point, which is absurd, being contrary to the nature of time. For though God could create

the universe ex nihilo with the appearance of age, even He could not create the world ex

nihilo actually having a past, to say nothing of an infinite past. But this is the only way

34

such a series of events could be formed, [if not by successive addition]. (Craig, 1979,

195)

This leaves only one alternative: the series of events in time was formed by successive addition.

The reason [why it is impossible to form an actual infinite by successive addition] is that

for every element one adds, one can always add one more. Therefore, one can never

arrive at infinity. What one constructs is a potential infinite only, an indefinite collection

that grows and grows as each new element is added. (Craig, 1979, 104)

Since Craig believes that there is no possible way that the PAI can be formed, he says the PAI

cannot exist. Here is my formal reconstruction of Craig’s argument:

1. If the PAI exists, it must have either been formed by “successive addition” or some

other way. [Premise]

2. The PAI cannot be formed by “successive addition.” [Premise]

3. The PAI cannot be formed by any way other than “successive addition” [Premise]

C. The PAI cannot exist. [1, 2, 3]

The problem in this argument is that the first Premise is false. It is not a necessary condition that

something be formed in order to exist. To seek the source of formation of the PAI is to

misunderstand the main idea of the PAI. The defining characteristic of the PAI is that it is not

formed. The PAI is a collection of events in time that extends infinitely far back into the past.

There is no moment where it was “formed.” This point can seem initially counterintuitive, but

grasping it is fundamental in understanding how the PAI can exist. Since Premise 1 is false, we

do not have reason to believe this argument’s conclusion that the PAI cannot exist.

35

ARGUMENT FROM ORDINAL NUMBERS

This argument functions as support for the second Premise of the Second Major Argument:


[Premise]

Before giving this argument, I will explain how it connects to the impossibility of the PAI. Using

this argument, Craig will conclude that the PAI would entail that the present moment in time is

separated from some moment in the past by an infinite number of other moments. In the

beginning of my paper, I gave two diagrams for how an actual infinite may be represented in

time. With this argument, Craig is referring to an actual infinite in time that takes another form:

Craig holds that if it were true that the present moment in time is separated from some moment

in the past by an infinite number of other moments, absurd conclusions would follow; therefore,

the PAI must not exist.

In this section, I will simply take for granted that it is true that “if the PAI entails that the

present moment in time is separated from some moment in the past by an infinite number of

36

other moments, then the PAI cannot exist” This is a dubious Premise, but not the one which I

will argue against here.10

Discussion of this argument will require us to become clear on the nature of ordinal

numbers. Roughly speaking, ordinal numbers describe the order of sets. We can begin by

thinking of ordinal numbers as they apply to finite sets. Imagine a race between three people.

The race’s judge takes down the name of each runner as they cross the finish line. We end up

with a set that looks like this:

{Ann, Bob, Carl}

Bob finished in 2nd place, so his ordinal number is 2. Bob has this particular ordinal number in

virtue of the relation under consideration between the members of this set. Let’s say instead that

we were ranking these individuals by height and the set came to look like this:

{Bob, Ann, Carl}

Now Bob is in 1st place, so his ordinal number is 1. This should highlight the importance of the

relation under consideration to the determination of a set’s ordinal number.

Additionally, if two sets are equivalent and are organized in accordance with the same

relation, then corresponding elements of each set share the same ordinal number. So, for

example, if there were another race between three people and the judge were to take down the

name of each runner as they cross the finish line, we could end up with a set that looks like this:

{Xian, Yanick, Zach}

10 I argue against this Premise in Infinite Divisibility of Time Argument.

37

This set is equivalent with the set that we analyzed for our first race (i.e., it has three elements)

and it is organized in accordance with the same relation as that of the set we analyzed for our

first race (i.e., order of runners crossing a finish line). From this we know that Yanick possesses

the same ordinal number as Bob, 2.

Up until this point we have been analyzing ordinal numbers as they apply to elements of

a set; however, ordinal numbers can also be used to refer to the order type of an entire set.

Reference to the ordinal number of sets rather than to the ordinal number of elements of sets still

conforms to similar rules as those that govern the ordinal number of elements of sets. There are

two important points that we must keep in mind when we consider the ordinal number of sets.

First, the ordinal number of a set is still given in accordance with the relation under consideration

for that set. Second, two sets with the same cardinal number also have the same ordinal number

if and only if they are organized in accordance with the same relation. We can observe that two

sets are organized in accordance with the same relation when there is a one-to-one mapping

between the two sets such that for any two members of the first set that are related by R there are

two members of the second set which are also related by R. To make this point more clear,

consider the two races once again:

{Ann, Bob, Carl} {Xian, Yanick, Zach}

If we map Ann to Xian, Bob to Yanick, and Carl to Zach, we can see that just as Ann came in

ahead of Bob, Xian came in ahead of Yanick; and just as Bob came in ahead of Carl, Yanick

came in ahead of Zach. Since these two sets have the same cardinal number and are organized in

accordance with the same relation, they have the same ordinal number. For a contrasting

example, imagine that in the second race, Yanick and Zach tied for 2nd place. Now the members

38

of the two sets are not organized in accordance with the same relation. Unlike the first example,

in this second example, the two sets do not have the same ordinal number.

The standard way of defining the ordinal number of a set is by reference to a paradigm.

The set of natural numbers serves as this paradigm; it possesses the ordinal number w*. Any set

which is equivalent to the set of natural numbers and is organized in accordance with the same

relation as the set of natural numbers has the ordinal number w*. The ordinal number w*+w* is

given to a set that is organized in such a way that the set is duplicated. So, while w* takes on the

form:

{1, 2, 3, …}

w*+w* takes on the form:

{1, 2, 3, …, 1, 2, 3, …}

With a better understanding of ordinal numbers, we can proceed to Craig’s presentation of the

argument:

Imagine a linearly arranged series of objects, say, balls, which is composed of an actually

infinite number of members … if there are aleph-null balls in the series, then they can be

ordinally numbered as we please: w*, w* + w*, or w* + w* + w* … for any collection

with aleph-null members in it can be ordered w* + w* … Hence a series of aleph-null

objects, if it could exist, would entail the existence of objects infinitely removed from any

given point. (Craig, 1991, 388)

39

In a set with aleph-null members, it can be alternatively assigned the ordinal number w* or

w*+w*. When it is organized as w* + w* we see that some elements of the set are separated by

an infinite number of elements:

{…, -3, -2, -1, …, -3, -2, -1}

If one were to count sequentially from the first -3 to the -3 that I have underlined above, one

would have to count an infinite number of elements before arriving to it. From this, Craig

concludes that a set with aleph-null members that can be alternatively assigned ordinal number

w* or w*+w* has some members that are separated by an infinite number of elements.


1. If a set has a cardinality of aleph-null, it can be assigned either the ordinal w* or the

ordinal w*+w*. [Premise]

2. The set of past infinite events has a cardinality of aleph-null. [Premise]

3. The set of past infinite events can be assigned the ordinal number w* + w*. [1, 2]

4. If the set of past infinite events can be assigned the ordinal number w* + w*, then that set

has some two elements that are separated from each other by an infinite number of

elements. [Premise]

C. The set of past infinite events has some two elements that are separated from each other

by an infinite number of events. [3, 4]

I will offer two arguments to refute this. The first argument attacks the first Premise. Craig’s

primary support for this Premise is his analogy to the infinite collection of balls. However, this

40

infinite collection of balls seems substantively different from collection of events in time. This

collection does not have a sequence to it; it is simply a collection of balls. This is why there

seems to be nothing unrepresentative about assigning it w* or w* + w*. To draw the parallel to

our analysis of ordinal numbers in finite sets, it seems that the relation under consideration for

the collection of balls is different from the relation under consideration for the series of events in

time. The series of events in time seems to have a meaningful sequential pattern. Each element

of the series came to exist after the end of another particular element of the series. In this way,

each element has a particular property which makes it unique from the other elements of the

series. This is unlike the collection of balls. In that collection, there is no such relevant

distinction between the elements, so any ordinal number appears to be arbitrary. This is what

allows us to say that the set could be alternatively assigned the ordinal number w* or w* + w*.

Since this arbitrariness does not appear to apply to all types of sets possessing the ordinal

number w*, the first Premise is untrue.

The second argument attacks the fifth Premise. In this Premise, Craig assumes that the

option of assigning a set the ordinal number w*+w* entails that the set has some elements which

are infinitely separated from each other. Craig is wrong about this assumption, however. While it

may be true that in a set possessing the ordinal number w*+w*, some members are infinitely

separated from each other, in a set possessing the ordinal number w*, no elements are infinitely

separated from each other. Consider the infinite set below:

{…, -3, -2, -1}

41

One could choose any two elements of this set and calculate exactly how many elements separate

the two. From this it is evident that there are no two elements of a set ordered w* which are

infinitely separated from each other.

Since the set ordered w* does not have any elements that are separated from each other

by an infinite number of elements, we can refute Craig’s argument by argument parity:

1. If a set has a cardinality of aleph-null, it can be assigned either the ordinal w* or the

ordinal w*+w* [Premise]

2. The set of past infinite events has a cardinality of aleph-null. [Premise]

3. The set of past infinite events can be assigned the ordinal number w*. [1, 2]

4. If the set of past infinite events can be assigned the ordinal number w*, no two elements

of the set are separated from each other by an infinite number of elements. [Premise]

C. The set of past infinite events does not have any two elements that are separated from

each other by an infinite number of events. [3, 4] (Notice that this conclusion is exactly

the opposite of the conclusion of Craig’s argument.)

We are now presented with two arguments which produce contradicting conclusions. Neither

argument is stronger than the other, so we have no reason to believe one conclusion over the

other. Therefore, Craig’s Argument from Ordinal Numbers has no impact on our reasons for

believing whether or not an actual infinite is possible.

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INFINITE DIVISIBILITY OF TIME ARGUMENT

In this section rather than argue against a specific argument that Craig makes, I will argue

against a Premise that underlies many arguments against the PAI. I briefly discussed this Premise

in the Argument from Ordinal Numbers section and mentioned that I believe it to be dubious.

The Premise is:

“If the PAI entails that the present moment in time is separated from some moment in the

past by an infinite number of other moments, then the PAI cannot exist.”

This Premise serves as the foundation for many arguments that Craig offers. Those arguments

take the following form:

- First, assert the Premise (“If the PAI entails that the present moment in time is

separated from some moment in the past by an infinite number of other moments,

then the PAI cannot exist.”).

- Then, argue that the PAI entails that the present moment in time is separated from

some moment in the past by an infinite number of other moments in time.

- Finally, conclude that the PAI could not exist.

I have not included all of Craig’s arguments which take this form. Instead, I took the Argument

from Ordinal Numbers to be representative of those arguments, since I believe it is the strongest

of those arguments. When I argued against the Argument from Ordinal Numbers, I did not refute

43

this Premise; however, in this section, I will. This section will have the added benefit of serving

as an alternate response to the Argument from Ordinal Numbers.

Craig’s justification for this Premise is that it would entail the problem of “traversing the

infinite.” Consider two moments in time, Moment A and Moment B. Imagine that between them

there exist an infinite number of other moments in time. Craig holds that we would never be able

to arrive at Moment B because we would have to “traverse” an infinite number of moments,

which is impossible. He says:

Suppose we imagine a man running through empty space on a path of stone slabs, a path

constructed such that when the man’s foot strikes the last slab, another appears

immediately in front of him. It is clear that even if the man runs for eternity, he will never

run across all the slabs. For every time his foot strikes the last slab, a new one appears in

front of him. The traditional cognomen for this is the impossibility of traversing the

infinite. (Craig, 1979, 104)

So, in short, since we cannot “traverse the infinite,” we cannot move between two points in time

if there are an infinite number of points between them. Craig says the PAI requires us to

“traverse the infinite”; therefore, the PAI must not exist. I argue that it is not impossible to

traverse the infinite. I will begin by stating my argument formally:

1. If it were impossible to “traverse the infinite,” it would be impossible to go from one

point in time to another point in time if there were an infinite number of points in time in

between them. [Premise]

2. Between any two points in time, there are an infinite number of points in time. [Premise]

44

3. If it were impossible to “traverse the infinite,” then it would be impossible to go from

any point in time to another point in time. [1, 2]

4. It is possible to go from a point in time to a later point in time. [Premise]

C. It is possible “traverse the infinite.” [3, 4]

The second Premise of my argument is the one needing the most argumentation. To understand it

more clearly, it may be useful to consider that it is equivalent to claiming that time is infinitely

divisible. Therefore, if I can show that time must be infinitely divisible, I will have proven the

second Premise. My strategy will be to demonstrate that if time were not infinitely divisible, we

would be left with an unacceptable absurdity.

If time were not infinitely divisible, that would mean that as we tried to divide it from

hours, to minutes, to seconds, to milliseconds, and so on, we would eventually reach a point

where it would be no longer possible to split it into smaller sections. We would be left with the

smallest and most basic unit of time possible. To illustrate my next example, let us assign a name

for this unit of time which cannot be divided: UT (Unit of Time). Now imagine two objects in

motion, Object A and Object B:

Object A moves at 1 mile per hour

Object B moves at 100 miles per hour.

The increment of time that we have chosen to gauge their speed is “hours.” However, let us

reduce this increment of time to UT. We will use x to indicate the distance that the object covers

over the course of UT. We can think of Object A as moving a pace of x per UT. Since Object B

moves at 100 times the speed of object A, object B moves at a pace of 100x per UT. If Object A

45

started its motion at a location demarcated as “point 1,” after one UT, object A would end at

“point 2.” On the other hand, if object B started at some location demarcated as “point 1,” then

after one UT, object B would end at “point 100.” This must be true given that Object B moves at

100 times the speed of Object A.

So far the absurdity is not yet clear, but imagine the implication of this scenario. At 1UT,

Object B is at point 1. At 2UT, object B is at point 100. Since UT is the most fundamental unit of

time, there cannot be any units of time in between 1UT and 2UT. But, in order for an object to

exist at a location, it must do so at a moment in time. This means that Object B could not have

existed at any of the points in between point 1 and point 100. It simply existed at point 1 at 1UT

and then point 100 at 2UT. In other words, while moving from point 1 to point 100, object B did

not cross points 2, 3, 7, 24, 52, 73, and so on. Instead, it simply existed at point 1 and then

somehow materialized at point 100. I will now construct this argument formally to make the

absurdity clear.

1. If time were finitely divisible, it would be possible for an object to move from one point to

another without “passing through” the intermediate points between them. [Premise]

2. In order to move from one point to another, an object must “pass through” the

intermediate points between them. [Premise]

C. Time cannot be finitely divisible. [1, 2]

The only way we can resolve this absurdity is if time were infinitely divisible. That way

there would be moments in time at which object B can exist at the intermediate points between

point 1 and point 100.

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Now that I have offered a defense of the second Premise of my argument, we can return

to my conclusion: “it is possible to ‘traverse the infinite.” Since it is possible to traverse the

infinite, we do not have reason to believe Craig’s crucial Premise Therefore we do not have

reason to believe Craig’s arguments which depend on this Premise.

CONCLUSION

William Lane Craig offers multiple arguments attempting to show a priori that the PAI is

impossible. In this paper, I have exposed the weakness in each of these arguments. Thus, we do

not have adequate reason to believe a priori that the PAI is impossible. This means that the

question of whether or not the past is infinite is one that turns on empirical evidence. We cannot

deny the possibility of an infinite past based on pure reason alone.

This conclusion also affects the Temporal Cosmological Argument. The Temporal

Cosmological Argument requires that there was a first event in time. Since we have shown that it

is possible that events extend infinitely far back into the past (i.e., there was no first event in

time), we have also shown that it is possible that the Temporal Cosmological Argument never

even gets off the ground. Therefore, we have shown the Temporal Cosmological Argument relies

on empirical evidence to confirm one of its fundamental premises.

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BIBLIOGRAPHY

Craig, William Lane and Walter Sinnott-Armstrong. 2004. God?: A Debate between a Christian

and an Atheist. Oxford: Oxford University Press.

Craig, William Lane and Quentin Smith. 1993. Theism, Atheism, and Big Bang Cosmology.

Oxford: Clarendon Press.

Craig, William Lane. 1991. "Time and Infinity." International Philosophical Quarterly 31.4:

387-401

Craig, William Lane. 1979. The Kalam Cosmological Argument. New York: Barnes & Noble

Books.

Craig, William Lane. 1980. “Philosophical and Scientific Pointers to Creatio Ex Nihilo.”

Perspectives on Science & Christian Faith: Journal of the American Scientific Affiliation

32 (5-13). Reprinted (1992) in R. Douglas Geivett and Brendan Sweetman, eds.,

Contemporary Perspectives on Religious Epistemology. New York: Oxford UP.

Melamed, Yitzhak. 2010. "Principle of Sufficient Reason." Stanford University. Stanford

University, <http://plato.stanford.edu/entries/sufficient-reason/>. Accessed on April 6th,

2014.

No Author. 2014a. Wikipedia. "Intuitionism." Wikimedia Foundation,

<http://en.wikipedia.org/wiki/Intuitionism>. Accessed on April 2nd, 2014.

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No Author. 2014b. Wikipedia. "Euclidean Geometry." Wikimedia Foundation,

<http://en.wikipedia.org/wiki/Euclidean_geometry>. Accessed on April 2nd, 2014.

Taylor, Richard. 1963. Metaphysics. Englewood Cliffs, NJ: Prentice-Hall.

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Defending the Possibility of an Infinite Past