An Endless Future is an Actual Infinite

An endless future is an actual infinite. Here’s how I’m using the terms:

• An actual infinite is a collection with infinitely many members. For present purposes, the relevant infinity is the first transfinite cardinal, ℵ_o. Thus, an actual infinite is a collection whose members can be put in one-to-one correspondence with the natural numbers. The members do not all need to (simultaneously) exist. They simply need to numerable or able to be numbered. (Craig, for instance, adopts presentism but still affirms that a beginningless past is an actual infinite. The members of this collection, qua past events, do not exist (under presentism); yet this is no barrier to their actual infinitude, for they are numerable and can be placed in one-to-one correspondence with the natural numbers.)
• A potential infinite is a collection that (i) always has finitely many members and (ii) continually increases (without limit) in its number of members.
• By ‘the future’, I mean everything later than the present, or everything that will occur/obtain/exist.
• Thus, letting T be a finite, non-zero, non-infinitesimal time interval (e.g., a day, second, or year), to say that the endless future is an actual infinite is to say that…
  • …the number of future Ts is ℵ_o (i.e., the members of the collection of future Ts can be put in one-to-one correspondence with the natural numbers); or, alternatively
  • …the number of Ts, each of which will occur, is ℵ_o (i.e., the members of the collection of every x such that x is a T and x will occur can be put in one-to-one correspondence with the natural numbers); or, alternatively
  • …the number of Ts later than the present T is ℵ_o (i.e., the members of the collection of every x such that x is a T and x is later than the present T can be put in one-to-one correspondence with the natural numbers).

Here are five arguments for the conclusion that an endless future is an actual infinite.

Argument #1

Suppose there’s an endless counter who counts a unique natural number per day of the endless future, beginning tomorrow with one. Since each day is paired with a unique natural number that the counter will count (and vice versa), it follows that the number of days in an endless future is the same as the number of natural numbers the counter will count. But the number of natural numbers the counter will count is aleph-null. This is because each natural number will eventually be counted at some point or other in the endless future – that is, there’s no number that the counter will fail to reach. So, since there are aleph-null-many natural numbers, and since each natural number will eventually be counted by the counter, it follows that the number of natural numbers the counter will count is aleph-null. Hence, since the number of days in an endless future is the same as that number, it follows that the number of days in an endless future is likewise aleph-null. Hence, an endless future involves an actually infinite collection of days – in which case, an endless future is an actual infinite.

Note that when we ask how many days there will be, we’re asking about the number of days later than the present day. We’re not asking about how many days there will have been between today and the ‘moving now’ as time progresses, or how many days will have been completed between today and the ‘moving now’ as time progresses. Yes, these are potential infinites. But that wasn’t the question. The question is posed in the simple future tense: how many events (or days) will occur. And if the future is endless, then aleph-null many days will occur. This doesn’t imply that there will ever come a time when infinitely many days have been completed, just as the fact that there are aleph-null-many numbers greater than 5 doesn’t imply that there’s some number infinitely greater than 5. But this doesn’t change the fact that each of infinitely many days is such that it will occur at some point in the endless future, just as each of infinitely many natural numbers is some finite distance from the number 5. The result is that an endless future is a collection with aleph-null-many elements. It involves aleph-null-many days, years, seconds, and events. Thus, it’s an actual infinite, not a potential infinite.

Argument #2

The second argument is an argument against the view that the endless future is a potential infinite. Here’s the argument:

1. If a collection is a potential infinite, then the number of its members increases over time. (Definitionally true)
2. So, if the collection of future years is a potential infinite, then the number of future years increases over time. (From 1)
3. The number of future years does not increase over time.
4. So, the collection of future years is not a potential infinite. (From 2, 3)

There are only two premises here. Premise (1) is definitionally true – a potential infinite is a collection that’s always finite but ever-increasing without limit. (2) trivially follows from (1), and (4) follows from (2) and (3). Thus, the only potentially controversial part of the argument is premise (3).

But premise (3) is clearly true. Consider the future years when 2023 is present:

When 2023 is present, the future years (i.e., the collection of years each of which will occur) is the collection {2024, 2025, 2026, 2027, …}. Now consider the future years when 2027 is present:

When 2027 is present, the future years is the collection {2028, 2029, 2030, 2031, …}. Notice that, as time progresses, years are removed from the collection of future years. The collection of future years when 2027 is present is a proper sub-collection of the collection of future years when 2023 is present.

Inspecting the diagrams, it’s evident that the number of future years does not increase over time. As time progresses, years are removed from the collection of future years. The collection does not increase over time. In fact, the collection of future years at any later time is always a proper sub-collection of the collection of future years at any earlier time. And when x is a proper sub-collection of y, x cannot have a more members than y. Hence, the number of future years does not increase over time. Premise (3) is true.

Thus, since (1)-(3) are true, (4) follows. The collection of future years is not a potential infinite.

Argument #3

Suppose it is now true that the number of days each of which will occur is just, say, 10. Then, time will come to an end in 10 days. This is because we are supposing that there will only be 10 more days in reality. So, in 10 days, time will come to an end. Hence, if time is endless – i.e., if time doesn’t come to an end – then it is false that the number of days each of which will occur is just 10.

But 10 was just an arbitrary finite number. Let the number be 1.8 quadrillion. The same argument applies: if it is now true that the number of days each of which will occur is just 1.8 quadrillion, then time will come to an end in 1.8 quadrillion days. Hence, if time is endless, then it is false that the number of days each of which will occur is 1.8 quadrillion. What this proves is that if the future is endless, the number of days each of which will occur is not any finite number. It must instead be actually infinite or aleph-null. For if it were a finite number – say, n – then time will come to an end n days from now. But time, we are supposing, is endless. Hence, it is not a finite number. It must, instead, be infinite.

Let’s put this formally. Letting n be an arbitrary finite number, we get:

1. If the number of days, each of which will occur, is n, then time comes to an end n days from now.

2. If time comes to an end n days from now, then time isn’t endless.
3. So, if time is endless, then the number of days, each of which will occur, is not n.

But n was just an arbitrary finite number. So, we can conclude to the generalized statement

4. So, if time is endless, then the number of days, each of which will occur, is not finite.

There are then two options for establishing actual infinitude from (4). The first option is to affirm that

5. If time is endless, then the number of days, each of which will occur, is at least 1.

We then add:

6. If the number of x’s is at least 1 but not any finite number, then the number of x’s is at least ℵ_o.

And from (4)–(6) it follows that

7. So, if time is endless, then the number of days, each of which will occur, is at least ℵ_o.

And that, of course, simply means that an endless future is an actual infinite.

The second way of proceeding adds to (4) the following two premises:

8. The number of future days is determinate.
9. If the number of x’s is determinate and not finite, then the number of x’s is infinite.

And from (4), (8), and (9) we get

10. So, if time is endless, then the number of days, each of which will occur, is infinite.

Once again, this means that an endless future is an actual infinite.

NB: Even open theists can accept premise (8). God can ensure (and presumably will ensure, if we take his promise of endless life seriously) that the future will be endless even though the precise content of that endless future is neither determined nor determinate.

Argument #4

If the future is endless, then the days later than today (i.e., the days that will occur) can be put in 1-to-1 correspondence with the natural numbers. This is easily seen: simply map tomorrow with 1, the next day with 2, the next day with 3, the next day with 4, and so on for each natural number. There is no natural number n such that n fails to get mapped to a unique future day. This entails that the number of days later than today is ℵ_o if the future is endless. Here’s the argument put formally:

1. If the future is endless, then the days later than today (i.e., the days that will occur) can be put in 1-to-1 correspondence with the natural numbers.

2. If the elements of a collection can be put in 1-to-1 correspondence with the natural numbers, then the number of elements in the collection is ℵ_o (i.e., actually infinite).
3. So, if the future is endless, then the number of days later than today (i.e., the days that will occur) is ℵ_o (i.e., actually infinite).

Premise (1) is clearly true: I just specified a procedure that provides the relevant one-to-one mapping between natural numbers and days later than today. To deny premise (1) requires that there is some natural number n such that n fails to be mapped to a unique future day. But there is no such number as n – it couldn’t be 20, since 20 is mapped to the future day 20 days from now; it can’t be 200, since 200 is mapped to the future day 200 days from now; it can’t be 2 quadrillion, since that’s mapped to the future day 2 quadrillion days from now; and so on. There simply can’t be a natural number n that fails to get mapped to a unique future day. And this means that every natural number is paired with a unique future day. And this, of course, means that premise (1) is true. Premise (2), moreover, is true by definition. Finally, the conclusion entails that an endless future involves an actually infinite collection of days. The endless future, in other words, is an actual infinite.

Argument #5

Mathematical induction is a deductively valid form of reasoning with the following structure, where ‘P(n)’ means ‘some predicate P is true of (natural number) n’ and where 1 is (stipulatively) the smallest natural number:

1. P(1). (Base case)
2. For each natural number n, if P(n), then P(n+1). (Inductive hypothesis)
3. So, for each natural number n, P(n).

Mathematical induction can be used to easily prove that if the future is endless, then the number of days each of which will occur is ℵ_o – in which case, the collection of future days is an actual infinite. Here’s how that proof goes:

Suppose the future is endless, and suppose Gabriel is going to count one natural number per day of the endless future, beginning with 1 and adding 1 each day. So, Gabriel will count 1 tomorrow, 2 the next day, 3 the next day, and so on. With this in hand, we can now run the mathematical induction:

1. It is true of natural number 1 that Gabriel will count 1. (Base case)

In premise (1), ‘P’ is the predicate ‘is a number that Gabriel will count’.

2. For each natural number n, if Gabriel will count n, then Gabriel will also count (n+1). (Inductive hypothesis)

Premise (2) is clearly true: if Gabriel eventually reaches natural number n, clearly Gabriel also eventually reaches (n+1), since Gabriel will simply count (n+1) the day after he counts n, and we’re supposing that Gabriel never stops counting one number per day. Thus, if Gabriel eventually reaches n but fails to reach (n+1), he would eventually stop counting, which contradicts our assumption that he never stops counting. Hence, it follows that if Gabriel will count n, then he will also count (n+1). And, of course, we’ve been speaking here in purely general terms, and so our conclusion here holds for each natural number n.

By mathematical induction, it follows that

3. So, for each natural number n, Gabriel will count n.

We now add:

4. There are ℵ_o-many natural numbers.
5. If (i) for each natural number n, Gabriel will count n, and (ii) there are ℵ_o-many natural numbers, then the number of natural numbers Gabriel will count is ℵ_o.

From (3)–(5) it follows that

6. So, the number of natural numbers Gabriel will count is ℵ_o.

Another premise:

7. If the number of natural numbers Gabriel will count is ℵ_o, then the collection of natural numbers Gabriel will count is actually infinite.

From (6) and (7) we get:

8. So, the collection of natural numbers Gabriel will count is actually infinite.

Another premise:

9. There’s a one-to-one correspondence between the collection of natural numbers Gabriel will count and the collection of future days within an endless future.

This just follows from the original setup: for each natural number Gabriel will count, there’s a unique future day on which Gabriel counts that number, and for each future day, there’s a unique natural number Gabriel will count on that day.

10.  If there’s a one-to-one correspondence between two collections, then they have the same number of elements.

From (8)–(10) we get

11. So, the collection of future days within the endless future is actually infinite.

And if the collection of future days within the endless future is actually infinite, then an endless future is an actual infinite. Hence, an endless future is an actual infinite.

Notice, also, that there is no illicit shift in this reasoning from each to all. I did not infer that there will ever come a time at which all future days have elapsed. Instead, the claim is that each of infinitely many natural numbers will be counted, and since there are infinitely many natural numbers and a one-to-one correspondence between the natural numbers and the days within an endless future, it follows that each of infinitely many days will occur. Hence, the number of days that will occur is ℵ_o. This is an actually infinite collection.

(NB: I address William Lane Craig’s objections to a similar, mathematical-induction-style argument in my video here.)

Concluding Notes

I don’t find these arguments terribly interesting, since it seems obvious to me that an endless future is an actual infinite. But there are many on the internet who, following Craig, think an endless future is only a potential infinite. These arguments are meant to help them see what I (take myself to) see. And the conclusion is significant: an endless future inherits the alleged “absurdities” attending actually infinite collections, such as having proper sub-collections equinumerous with their whole collections and (allegedly) resisting our intuitive notions of addition and subtraction. I explain this in depth in my video on Hilbert’s Hotel for those interested.

Three final notes. First, I didn’t overlook tensed views of time; my premises evidently do not assume a tenseless view of time. Even if time is tensed, the number of days that will occur if the future is endless is ℵ_o. For more on this point about tense (and for more arguments for the conclusion that an endless future is an actual infinite), see the excellent Malpass and Morriston paper here as well as my Hilbert’s Hotel video mentioned earlier

Second, Josh Rasmussen has suggested that, under some open future views, one can hold that while there will always be more days, it is undetermined which days those will be. In that case, it could be that there is presently a last day d that is entailed by present truths, and hence that there is no day in particular after d that is entailed by present truths. This is an intriguing possibility, but I don’t see how it would rebut the arguments I’ve given. Even if there is no day in particular after d entailed by present truths, it’s still determinately true that some day or other is after d, and some day or other is after that, and some day or other is after that, and so on ad infinitum. There’s a determinate number of future days here (to wit, ℵ_o), even though the precise character of those days is indeterminate. (I’m assuming, of course, that it is determinately true that the future is endless in the sense that it is determinately false that there will be some day d* such that there will be no days later than d*.)

Third, I don’t claim that the five arguments are all independent from one another. My goal here is simply to help people see what I see.