J.P. Moreland offers the following argument against a beginningless past in Scaling the Secular City:

"It is impossible to count to infinity. For if one counts forever and ever, he will still be, at every moment, in a place where he can always specify the number he is currently counting. Furthermore, he can always add one more member to what he has counted and thereby increase the series by one. A series formed by successive addition is a potential infinite. Such a series can increase forever without limit, but it will always be finite. This means that the past must have been finite. For the present moment is the last member of the series of past events formed by successive addition. And since one cannot reach infinity one at a time, then if the past was actually infinite, the present moment could not have been reached. For to come to the present moment, an actual infinite would have to have been crossed." (p. 29, my copy. Italics added)

We can express the argument a bit more formally as follows:

1. At every point in the growth of any potential infinite, one can specify its cardinal number via a natural number and increase that number by 1. (premise)

2. If at every point in the growth of any potential infinite, one can specify its cardinal number via a natural number and increase that number by 1, then no potential infinite can be transformed into an actual infinite. (premise)

3. Therefore, no potential infinite can be transformed into an actual infinite. (from 1 and 2)

4. Any series formed by successive addition is a potential infinite. (premise)

5. The past is a series formed by successive addition. (premise)

6. Therefore, the past is a potential infinite. (from 4 and 5)

7. Therefore, the past cannot be actually infinite. (from 3 and 6)

The argument is thus formally valid; so if the premises should turn out to be true, then the conclusion follows of necessity. Why, then, should we accept the premises?

Well, as is indicated above, (3) follows (1) and (2), (6) follows from (4) and (5), and (7) follows from (3) and (6), Furthermore, (5) is beyond dispute, and I'm inclined to accept (1) and (2). That leaves (4), i.e., the claim that any series formed by successive addition is a potential infinite. Why should we accept it?

Now in the passage above, Moreland offers a good reason for thinking that any series formed by successive addition that has a beginning is a potential infinite -- i.e., any series naturally represented by the following series of natural numbers:

1 2 3 ....

Furthermore, he reasoned persuasively that no such series can be transformed from a potential infinite into an actual infinite.

However, that line of reasoning has no immediate bearing against the prospects of a beginningless series formed by successive addition, i.e., a series naturally represented by the following series of negative integers:

...-3 -2 -1

Rather, all that follows from that line of reasoning is the weaker claim that if the latter series is possible, it doesn't involve the transformation of a potential infinite into an actual infinite. But of course, those not antecedently convinced of the claimed necessary finitude of the past agree with that. For if the past should turn out to be beginningless, then some infinite set of events or other exists at each point in the past. But if so, then there is no event in the past that involved going from a state of not having traversed at least one infinite set of events to having traversed at least one such set. And if that's right, then if a beginningless past is possible, then it is a series of events formed by successive addition that does not involve turning a potential infinite into an actual infinite.

Thus, those who are antecedently unconvinced of the necessity of a beginningless past believe that it's at least epistemically possible that (i) the past is actually infinite, (ii) it was formed by successive addition, and (iii) the formation of the past did not involve turning a potential infinite into an actual infinite. But if a past of this sort should turn out to be possible, premise (4) is false. Thus, to adequately support premise (4), Moreland must come up with a line of reasoning that rules out the epistemic possibility expressed by (i)-(iii). But as we've seen, Moreland's reasoning in the above-quoted argument fails to do that; rather, it only rules out the possibility of an actually infinite series formed by successive addition that has a beginning. I conclude, then, that Moreland's grounds for premise (4) are inadequate.

Perhaps, though, Moreland construes a beginningless past in the way he does in an attempt to be charitable? For one might worry that if, in a beginningless past, some infinite set or other is traversed before every event, then such a past has at least one infinite proper subset of events that wasn’t formed by successive addition, which is absurd.

I don't know if this is why Moreland construes a beginningless past in the way he does, but such a worry would be ill-founded, based as it is on an inference involving an instance of the quantifier shift fallacy, reasoning from

1) Every point in a beginningless past is such that there exists an actual infinite set of events that existed prior to it.

to

2) There is an actual infinite set of events, such that it exists prior to every point in a beginningless past.

which is the same illicit pattern of inference involved in reasoning that because every child has a mother who directly gave birth to them, there is a mother who directly gave birth to every child.

No, if the past is beginningless, then while an infinite subset of events exists prior to each event, it's a new infinite every time. To illustrate: pick any event --say, the present day -- and represent it by the integer -1. Then the set of past days traversed for each of the previous days, and including today, can be represented as follows:

...

....

.....

2 days ago: {..., -5, -4, -3}

1 day ago: {..., -5, -4, -3, -2}

Present day: {..., -5, -4, -3, -2, -1}

Thus, if a past of this sort is possible, then as is represented above, the set of days traversed at each day of the past is actually infinite. However, at each day, the set of days traversed is different. So, for example, the set of days traversed today contains, in addition to the set of days traversed yesterday, the new member represented by -1, viz., today. Thus, as I said a moment ago, if the past is beginningless, then while the set of events traversed at each point in the past is actually infinite, it's a new set every time, as each passing event adds a new member to the previous set.

"It is impossible to count to infinity. For if one counts forever and ever, he will still be, at every moment, in a place where he can always specify the number he is currently counting. Furthermore, he can always add one more member to what he has counted and thereby increase the series by one. A series formed by successive addition is a potential infinite. Such a series can increase forever without limit, but it will always be finite. This means that the past must have been finite. For the present moment is the last member of the series of past events formed by successive addition. And since one cannot reach infinity one at a time, then if the past was actually infinite, the present moment could not have been reached. For to come to the present moment, an actual infinite would have to have been crossed." (p. 29, my copy. Italics added)

We can express the argument a bit more formally as follows:

1. At every point in the growth of any potential infinite, one can specify its cardinal number via a natural number and increase that number by 1. (premise)

2. If at every point in the growth of any potential infinite, one can specify its cardinal number via a natural number and increase that number by 1, then no potential infinite can be transformed into an actual infinite. (premise)

3. Therefore, no potential infinite can be transformed into an actual infinite. (from 1 and 2)

4. Any series formed by successive addition is a potential infinite. (premise)

5. The past is a series formed by successive addition. (premise)

6. Therefore, the past is a potential infinite. (from 4 and 5)

7. Therefore, the past cannot be actually infinite. (from 3 and 6)

The argument is thus formally valid; so if the premises should turn out to be true, then the conclusion follows of necessity. Why, then, should we accept the premises?

Well, as is indicated above, (3) follows (1) and (2), (6) follows from (4) and (5), and (7) follows from (3) and (6), Furthermore, (5) is beyond dispute, and I'm inclined to accept (1) and (2). That leaves (4), i.e., the claim that any series formed by successive addition is a potential infinite. Why should we accept it?

Now in the passage above, Moreland offers a good reason for thinking that any series formed by successive addition that has a beginning is a potential infinite -- i.e., any series naturally represented by the following series of natural numbers:

1 2 3 ....

Furthermore, he reasoned persuasively that no such series can be transformed from a potential infinite into an actual infinite.

However, that line of reasoning has no immediate bearing against the prospects of a beginningless series formed by successive addition, i.e., a series naturally represented by the following series of negative integers:

...-3 -2 -1

Rather, all that follows from that line of reasoning is the weaker claim that if the latter series is possible, it doesn't involve the transformation of a potential infinite into an actual infinite. But of course, those not antecedently convinced of the claimed necessary finitude of the past agree with that. For if the past should turn out to be beginningless, then some infinite set of events or other exists at each point in the past. But if so, then there is no event in the past that involved going from a state of not having traversed at least one infinite set of events to having traversed at least one such set. And if that's right, then if a beginningless past is possible, then it is a series of events formed by successive addition that does not involve turning a potential infinite into an actual infinite.

Thus, those who are antecedently unconvinced of the necessity of a beginningless past believe that it's at least epistemically possible that (i) the past is actually infinite, (ii) it was formed by successive addition, and (iii) the formation of the past did not involve turning a potential infinite into an actual infinite. But if a past of this sort should turn out to be possible, premise (4) is false. Thus, to adequately support premise (4), Moreland must come up with a line of reasoning that rules out the epistemic possibility expressed by (i)-(iii). But as we've seen, Moreland's reasoning in the above-quoted argument fails to do that; rather, it only rules out the possibility of an actually infinite series formed by successive addition that has a beginning. I conclude, then, that Moreland's grounds for premise (4) are inadequate.

Perhaps, though, Moreland construes a beginningless past in the way he does in an attempt to be charitable? For one might worry that if, in a beginningless past, some infinite set or other is traversed before every event, then such a past has at least one infinite proper subset of events that wasn’t formed by successive addition, which is absurd.

I don't know if this is why Moreland construes a beginningless past in the way he does, but such a worry would be ill-founded, based as it is on an inference involving an instance of the quantifier shift fallacy, reasoning from

1) Every point in a beginningless past is such that there exists an actual infinite set of events that existed prior to it.

to

2) There is an actual infinite set of events, such that it exists prior to every point in a beginningless past.

which is the same illicit pattern of inference involved in reasoning that because every child has a mother who directly gave birth to them, there is a mother who directly gave birth to every child.

No, if the past is beginningless, then while an infinite subset of events exists prior to each event, it's a new infinite every time. To illustrate: pick any event --say, the present day -- and represent it by the integer -1. Then the set of past days traversed for each of the previous days, and including today, can be represented as follows:

...

....

.....

2 days ago: {..., -5, -4, -3}

1 day ago: {..., -5, -4, -3, -2}

Present day: {..., -5, -4, -3, -2, -1}

Thus, if a past of this sort is possible, then as is represented above, the set of days traversed at each day of the past is actually infinite. However, at each day, the set of days traversed is different. So, for example, the set of days traversed today contains, in addition to the set of days traversed yesterday, the new member represented by -1, viz., today. Thus, as I said a moment ago, if the past is beginningless, then while the set of events traversed at each point in the past is actually infinite, it's a new set every time, as each passing event adds a new member to the previous set.

## Comments

In your critique of J.P's argument in this section you said,

Moreland offered a good reason for thinking that any series formed by successive additionthat has a beginningis a potential infiniteWhat J.P Said is,

"A series formed by successive addition is a potential infinite. Such a series can increase forever without limit, but it will always be finite."He didn't say exactly what you said. You assume he began with the premise of a limitation. He actually demonstrated that anything that brings us to a definitive present point must by THAT DEMONSTRATION be also finite. This is an evidentury finding not a presupposition that YOU seem to place on his findings.

Further, 4 would be the only conclusion based on the acceptance of 2 which I believe is established and makes sense.

I believe your critique is more than wishful thinking. To invoke tranversing of events across an infinite past would make all events (transversals also)finite with a ending point of the present...To me your's is a TOTALLY self defeating argument...and you say something like it's

epistemically possible???How is that?

Then you said this:

"No, if the past is beginningless, then while an infinite subset of events exists prior to each event,it's a new infinite every time, in which case no such subset failed to be formed by successive addition"Wouldn't saying

"A New infinite"be the same thing as saying that it was FINITE?Wow!

You wrote:

In your critique of J.P's argument in this section you said,Moreland offered a good reason for thinking that any series formed by successive addition that has a beginning is a potential infinite

What J.P Said is, "A series formed by successive addition is a potential infinite. Such a series can increase forever without limit, but it will always be finite."

He didn't say exactly what you said.

Right, he didn't. My point was that although he asserts that all series formed by successive addition are potential infinites, the reasoning he offers in the quoted argument only supports the qualified claim that all series

with beginningsformed by successive addition are potential infinites.You wrote:

Further, 4 would be the only conclusion based on the acceptance of 2 which I believe is established and makes sense.Well, I argued that it isn't the only conclusion. Perhaps you could point out a mistake in my reasoning?

You wrote:

I believe your critique is more than wishful thinking. To invoke tranversing of events across an infinite past would make all events (transversals also)finite with a ending point of the present.

...To me your's is a TOTALLY self defeating argument...and you say something like it's epistemically possible???

How is that?

I'm afraid I didn't follow that. Perhaps you can rephrase your reasoning?

You wrote:

Then you said this:"No, if the past is beginningless, then while an infinite subset of events exists prior to each event, it's a new infinite every time, in which case no such subset failed to be formed by successive addition"

Wouldn't saying "A New infinite" be the same thing as saying that it was FINITE?

No, it wouldn't.

Consider the following two series:

A: ...-5 -4 -3

B: ...-4 -3 -2

Series A and B are both actually infinite. Furthermore, if one traversed from A to B, then one would traverse to a new actual infinite. For the set

B: {...-4, -3, -2}

contains a member that A does not, viz. -2. So saying "a new infinite" is not saying the same thing as saying that the prior set -- in this case, A -- was finite.

Best,

EA

Using their reasoning, we can easily show that it is impossible for time to progress at all, because there are infinitely many moments between every moment that they care to define. And yet, time progresses anyway....