(The rest of the posts in this series can be found here.)

In some of his popular apologetics writings, Craig uses the following argument to support a key premise in his kalam cosmological argument, viz., that the universe began to exist:

"Suppose we meet a man who claims to have been counting down from infinity and who is now finishing: . . ., -3, -2, -1, 0. We could ask, why didn’t he finish counting yesterday or the day before or the year before? By then an infinite time had already elapsed, so that he should already have finished. Thus, at no point in the infinite past could we ever find the man finishing his countdown, for by that point he should already be done! In fact, no matter how far back into the past we go, we can never find the man counting at all, for at any point we reach he will already have finished. But if at no point in the past do we find him counting, this contradicts the hypothesis that he has been counting from eternity. This shows again that the formation of an actual infinite by never beginning but reaching an end is as impossible as beginning at a point and trying to reach infinity."

Call this "the immortal counter" argument". The argument can be expressed as a reductio, with (1) below as the premise set up for reduction:

1. The past is beginningless (conceived as a set of events with the cardinality A0, and the order-type w*).

2. If the past is beginningless, then there could have been an immortal counter who counts down from such a past at the rate of one negative integer per day.

3. The immortal counter will finish counting if and only if he has an infinite number of days in which to count them.

4. If the past is beginningless, then there are an infinite number of days before every day.

5. Therefore, the immortal counter will have finished counting before every day.

6. If the immortal counter will have finished counting before every day, then he has never counted.

7. Therefore, the immortal counter has both never counted and has been counting down from a beginningless past (contradiction)

8. Therefore, the past is not beginningless (from 1-7, reductio).

The undercutting defeater can be brought out by a careful look at (3). Grant the 'only if'. But why think the immortal counter will finish his count if he has had an infinite number of days to count them? For it's epistemically possible that he's counted down an infinite number of negative integers from a beginningless past, and yet has not counted them all. So, for example, he could now be counting "-3", so that he has just finished counting an infinite number of negative integers, viz., {...-5, -4, -3}, and yet he has not counted down all the negative integers. Given this epistemic possibility, any reason for believing his (3) is undercut.

Craig thinks he has a reply to this.

“I do not think the argument makes this alleged equivocation [from 'infinite' to 'all'. -EA], and this can be made clear by examining the reason why our eternal counter is supposedly able to complete a count of the negative numbers, ending at zero. In order to justify this intuitively impossible feat, the argument’s opponent appeals to the so-called Principle of Correspondence…On the basis of the principle the objector argues that since the set of past years can be put into a one-to-one correspondence with the set of negative numbers, it follows that by counting one number a year an eternal counter could complete a countdown of the negative numbers by the present year. If we were to ask why the counter would not finish next year or in a hundred years, the objector would respond that prior to the present year an infinite number of years will have elapsed, so that by the Principle of Correspondence, all the numbers should have been counted by now.

But this reasoning backfires on the objector: for on this account the counter should at any point in the past have already finished counting all the numbers, since a one-to-one correspondence exists between the years of the past and the negative numbers.” (Craig, “Review of Time, Creation, and the Continuum”, p. 323.)

Thus, Craig thinks the objector is committed to the claim that the counter will finish his count iff the days he's counted can be put into a 1-1 correspondence with the set of natural numbers. And since this can be done at any day of a beginningless past, the counter should always be done. But that contradicts the hypothesis that he's been counting down from a beginningless past.

But this won't do at all. For why, exactly, must the objector presuppose that the counter will finish his count iff the set of days he counts can be put into 1-1 correspondence with the set of natural numbers? Craig says that it's because otherwise the objector can't account for the possibility of an immortal counter who finishes the task on a particular day, as opposed to any other day. Now granted, counting a set of days that can be put into such a correspondence is a necessary condition for counting down a beginningless set of negative integers, but why in the world are we supposed to think it also sufficient?

Call the biconditional above 'Craig's Claim' (hereafter 'CC'):

(CC) The counter will have finished counting all of the negative integers if and only if the years of the past can be put into a one-to-one correspondence with them.

Now consider the following epistemically possible scenario as an undercutting defeater for CC:

Suppose God timelessly numbers the years to come about in a beginningless universe. Suppose further that He assigns the negative integers to the set of events prior to the birth of Christ, and then the positive integers begin at this point. Then the timeline, with its corresponding integer assignment, can be illustrated as follows:

…-3 -2 -1 Birth of Christ 1 2 3…

Suppose yet further that God assigned Ralph, an immortal creature, the task of counting down the negative integers assigned to the years BCE, and stopping at the birth of Christ. Call this task ‘T’. With this in mind, suppose now that Ralph has been counting down from eternity past and is now counting the day assigned (by God) the integer -3. In such a case, Ralph has counted a set of years that could be put into a one-to-one correspondence with the set of negative integers, yet he has not finished all the negative integers.

This case shows that, while it is a necessary condition for counting all of the events that one is able to put them into a one-to-one correspondence with the natural numbers, we have reason to doubt that it's sufficient. For if the events that are to be counted have independently “fixed”, or, “designated” integer assignments set out for one to traverse, one must count through these such that, for each event, the number one is counting is the same as the one independently assigned to the event. In the scenario mentioned above, God assigned an integer to each year that will come to pass. In such a case, Ralph must satisfy at least two conditions if he is to accomplish T: (i) count a set of years that can be put into a one-to-one correspondence with the natural numbers, and (ii) for each year that elapses, count the particular negative integer that God has independently assigned to it. According to CC, however, Ralph is supposed to be able to accomplish T by satisfying (i) alone. But we have just seen that he must accomplish (ii) as well. Therefore, being able to place the events of the past into a one-to-one correspondence with the natural numbers does not guarantee that the counter has finished the task of counting all the negative integers. And given that this scenario is epistemically possible CC is undercut. But recall that CC is Craig’s rationale for (3). Thus, (3) is undercut.

In some of his popular apologetics writings, Craig uses the following argument to support a key premise in his kalam cosmological argument, viz., that the universe began to exist:

"Suppose we meet a man who claims to have been counting down from infinity and who is now finishing: . . ., -3, -2, -1, 0. We could ask, why didn’t he finish counting yesterday or the day before or the year before? By then an infinite time had already elapsed, so that he should already have finished. Thus, at no point in the infinite past could we ever find the man finishing his countdown, for by that point he should already be done! In fact, no matter how far back into the past we go, we can never find the man counting at all, for at any point we reach he will already have finished. But if at no point in the past do we find him counting, this contradicts the hypothesis that he has been counting from eternity. This shows again that the formation of an actual infinite by never beginning but reaching an end is as impossible as beginning at a point and trying to reach infinity."

Call this "the immortal counter" argument". The argument can be expressed as a reductio, with (1) below as the premise set up for reduction:

1. The past is beginningless (conceived as a set of events with the cardinality A0, and the order-type w*).

2. If the past is beginningless, then there could have been an immortal counter who counts down from such a past at the rate of one negative integer per day.

3. The immortal counter will finish counting if and only if he has an infinite number of days in which to count them.

4. If the past is beginningless, then there are an infinite number of days before every day.

5. Therefore, the immortal counter will have finished counting before every day.

6. If the immortal counter will have finished counting before every day, then he has never counted.

7. Therefore, the immortal counter has both never counted and has been counting down from a beginningless past (contradiction)

8. Therefore, the past is not beginningless (from 1-7, reductio).

The undercutting defeater can be brought out by a careful look at (3). Grant the 'only if'. But why think the immortal counter will finish his count if he has had an infinite number of days to count them? For it's epistemically possible that he's counted down an infinite number of negative integers from a beginningless past, and yet has not counted them all. So, for example, he could now be counting "-3", so that he has just finished counting an infinite number of negative integers, viz., {...-5, -4, -3}, and yet he has not counted down all the negative integers. Given this epistemic possibility, any reason for believing his (3) is undercut.

Craig thinks he has a reply to this.

“I do not think the argument makes this alleged equivocation [from 'infinite' to 'all'. -EA], and this can be made clear by examining the reason why our eternal counter is supposedly able to complete a count of the negative numbers, ending at zero. In order to justify this intuitively impossible feat, the argument’s opponent appeals to the so-called Principle of Correspondence…On the basis of the principle the objector argues that since the set of past years can be put into a one-to-one correspondence with the set of negative numbers, it follows that by counting one number a year an eternal counter could complete a countdown of the negative numbers by the present year. If we were to ask why the counter would not finish next year or in a hundred years, the objector would respond that prior to the present year an infinite number of years will have elapsed, so that by the Principle of Correspondence, all the numbers should have been counted by now.

But this reasoning backfires on the objector: for on this account the counter should at any point in the past have already finished counting all the numbers, since a one-to-one correspondence exists between the years of the past and the negative numbers.” (Craig, “Review of Time, Creation, and the Continuum”, p. 323.)

Thus, Craig thinks the objector is committed to the claim that the counter will finish his count iff the days he's counted can be put into a 1-1 correspondence with the set of natural numbers. And since this can be done at any day of a beginningless past, the counter should always be done. But that contradicts the hypothesis that he's been counting down from a beginningless past.

But this won't do at all. For why, exactly, must the objector presuppose that the counter will finish his count iff the set of days he counts can be put into 1-1 correspondence with the set of natural numbers? Craig says that it's because otherwise the objector can't account for the possibility of an immortal counter who finishes the task on a particular day, as opposed to any other day. Now granted, counting a set of days that can be put into such a correspondence is a necessary condition for counting down a beginningless set of negative integers, but why in the world are we supposed to think it also sufficient?

Call the biconditional above 'Craig's Claim' (hereafter 'CC'):

(CC) The counter will have finished counting all of the negative integers if and only if the years of the past can be put into a one-to-one correspondence with them.

Now consider the following epistemically possible scenario as an undercutting defeater for CC:

Suppose God timelessly numbers the years to come about in a beginningless universe. Suppose further that He assigns the negative integers to the set of events prior to the birth of Christ, and then the positive integers begin at this point. Then the timeline, with its corresponding integer assignment, can be illustrated as follows:

…-3 -2 -1 Birth of Christ 1 2 3…

Suppose yet further that God assigned Ralph, an immortal creature, the task of counting down the negative integers assigned to the years BCE, and stopping at the birth of Christ. Call this task ‘T’. With this in mind, suppose now that Ralph has been counting down from eternity past and is now counting the day assigned (by God) the integer -3. In such a case, Ralph has counted a set of years that could be put into a one-to-one correspondence with the set of negative integers, yet he has not finished all the negative integers.

This case shows that, while it is a necessary condition for counting all of the events that one is able to put them into a one-to-one correspondence with the natural numbers, we have reason to doubt that it's sufficient. For if the events that are to be counted have independently “fixed”, or, “designated” integer assignments set out for one to traverse, one must count through these such that, for each event, the number one is counting is the same as the one independently assigned to the event. In the scenario mentioned above, God assigned an integer to each year that will come to pass. In such a case, Ralph must satisfy at least two conditions if he is to accomplish T: (i) count a set of years that can be put into a one-to-one correspondence with the natural numbers, and (ii) for each year that elapses, count the particular negative integer that God has independently assigned to it. According to CC, however, Ralph is supposed to be able to accomplish T by satisfying (i) alone. But we have just seen that he must accomplish (ii) as well. Therefore, being able to place the events of the past into a one-to-one correspondence with the natural numbers does not guarantee that the counter has finished the task of counting all the negative integers. And given that this scenario is epistemically possible CC is undercut. But recall that CC is Craig’s rationale for (3). Thus, (3) is undercut.

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