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
*T*s is ℵ_{o}(i.e., the members of the collection of future*T*s can be put in one-to-one correspondence with the natural numbers); or, alternatively - …the number of
*T*s, 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
*T*s*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).

- …the number of future

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:

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

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

- If time comes to an end
*n*days from now, then time isn’t endless. - 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

- 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

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

We then add:

- 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

- 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:

- The number of future days is determinate.
- 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

- 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:

- 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.

- 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). - 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:

- P(1). (Base case)
- For each natural number
*n*, if P(*n*), then P(*n*+1). (Inductive hypothesis) - 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:

- 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’.

- 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

- So, for each natural number
*n*, Gabriel will count*n*.

We now add:

- There are ℵ
_{o}-many natural numbers. - 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

- So, the number of natural numbers Gabriel will count is ℵ
_{o}.

Another premise:

- 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:

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

Another premise:

- 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.

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

From (8)–(10) we get

- 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.

These arguments nicely advance the discussion.

I continue to be curious about an idea I floated by email. Basically, a

dynamic-futuremodel may open up a different analysis. On this model, unlike the fixity of the past, there is no fixed number of future events. (This model probably works best on variable domain semantics, where different times and worlds have different domains of quantification.)Here is how the dynamic model might combine endless time with a finite future. Suppose there is a certain, finite number of future days determined by present states. Still, that number is changeable. For example, suppose there are 10^55 days left before this universe enters a concluding event, C. Perhaps new days beyond C could be added (based on new information in the present) before C arrives. If new days are continually added to the stack of future days, then even if there are always only finite future days, perhaps there will never be a time after which there are no more days. In that sense, time would still count as endless.

It is perhaps easy to see how this model might affect the five arguments. Suppose the future is not entirely fixed or filled in (already there). Then, regarding the first argument, while a counter may be set to keep counting, it might not be true that for every natural number, there is (“already”) a future day corresponding to the counting of number. Instead, there might be this truth in the neighborhood: the counter will count whatever future days are on the stack, and since days keep getting added to the stack, there will never be a day when the stack of future days is empty.

Regarding the second argument, the dynamic model predicts that the number of future days is variable. So, it might grow (as new grounds of future states comes into being) or shrink (as fewer grounds of the future may be available — e.g., near the end of a cycle).

Regarding argument three, one might question (1) or (2), depending on the precise meaning of “time comes to an end.” According to a dynamic future, there is a final day, D, and in that sense we could say time *presently* comes to an end at D. But if the future is variable, then when D arrives, it will no longer be the case that time comes to an end at D. Instead, time’s life will have been extended to at least another day. 🙂

Similar idea for the other arguments: there is a last day on the stack of future days, but that day changes before it is reached. Applied to the inductive argument, either the inductive step is not true (because it purports to quantify over a domain that doesn’t yet entirely exist) or it is to be interpreted as describing a variable domain — e.g., that if G counts n, and IF day n+1 is on the stack, G will count n + 1. Etc.

A potential lesson: if this analysis is on the right track, then your arguments have a certain additional value. They help us see an unhappy mixture of these three views: (i) finitism, (ii) endless time, and (iii) fixity of the future. That’s significant in no small part because (iii) would seem to be required for certain theories of foreknowledge, including a theory held by some who also hold (i) and (ii). Seeing this is progress for all of us.

I hope some of these notes serve the inquiry.

Thanks for the comment, Josh!

I’m still working through this ‘Variable Domain Open Future (VDOF)’ proposal in my own mind. One thing I’m certain about is that VDOF serves the inquiry 🙂

One option is to grant that the arguments require ~VDOF as a background assumption. As you point out, this is significant. Non-open theists who accept an endless future (e.g., Craig and co.) are committed to ~VDOF, and so the arguments create a real bind for them if they want to mount the WLC-style Kalam. If the arguments really do require ~VDOF, then proponents of the WLC-style Kalam are, I think, committed to (a certain kind of) open theism. (Assuming, of course, that they think the future could be endless.) Very significant indeed!

Just saw this: “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.”

The semantics are tricky to my mind, but with a variable domain, I don’t see that the consequent is required. Maybe there’s a semantics that could help here. There’s certainly more to explore…

I think I’m stuck on the variables of quantification. They aren’t indeterminate objects if the domain has only determinate objects.

I suppose one could keep constructing a longer and longer sentence to express an increasingly complex proposition with more clauses. But of course, a finitist doesn’t think any sentence is actually infinitely long, or that any proposition is infinitely complex. More to the point, an *dynamic futurist* theorist doesn’t think there’s infinitely truthmakers (or events) already laid out into the future. So at some level of complexity, the sentence will express a proposition that isn’t true — doesn’t correspond to anything.

I don’t see a way around that. Presently. 🙂

I guess my thought was that while the finitist doesn’t think that there is any such sentence or proposition, God’s promise to give us endless life may imply that there is. Like, in order for God’s promise to be true (or a successful speech act), it needs to be true that there will be a day tomorrow, and then a day after

that, and then a day afterthat, and so onad infinitum. But there may be moves around this — cf. my previous comment. Once more, I’m not sure! 🙂I guess my thought was that while that consequent may not be required by variable domain semantics, it seems required by God’s promise of unending life. (In line with Morriston, Malpass, and co., I’m mainly running this argument as an ad hominem against those who think God grants us unending life.) I was thinking that this promise (reflecting an infallible divine intention, thus giving us a present truth that determines future facts) implies that it is now determinately true that there will be a day tomorrow, and some day or other after that, and some day or other after

that, and some day or other afterthat, and so onad infinitum.Perhaps God’s intentions (about the duration of the future) are

alsovariable? Like, God’s current intentions only determine a final day some finite distance in the future, but later on God will intend still further days. (But then one might wonder: is it reallytrue,right now, that God intends anendlessafterlife? His current intentions, under this proposal, are compatible with the future really coming to an end in a finite time, it seems.)Or perhaps I’m mistaken to treat it as Christian data that God really does intend (or really has intended) to give us an endless afterlife. Like, maybe his intentions really only ‘cover’ finitely many future days, but he’ll continually (never-endingly) generate new intentions of this sort. One way to fill this out is to say that perhaps God’s promise of an endless afterlife is just an expression of his commitment to keep on generating such intentions.

I don’t know. This is all on the edge of my thinking!

Regarding Joe’s first argument, that an endless future entails an actual infinite, specifically, an endless number of days. However, this assumption can be challenged. You could argue that while there is no end to a future time, it does not follow that there are an infinite number of days. Rather, the future could consist of a potentially infinite, which is collection that is always finite but continually increasing in its number of members without limit.

Moreover, even if one grants the assumption that the future involves an actual infinite, it is not clear that the argument shows why an endless future must be an actual infinite. The argument assumes that the number of days in an endless future is the same as the number of natural numbers, but this is not necessarily the case. One could argue that time is not like natural numbers, and thus, there is no one-to-one correspondence between time and numbers.

Regarding the second argument, it is true that a potential infinite is a collection that continually increases without limit. However, it is not clear that this condition is a necessary condition for a potential infinite. You could argue that a potential infinite is simply a collection that is always finite, even if it does not continually increase.

Furthermore, the argument assumes that the collection of future days does not grow, but this is not necessarily the case. You could argue that the collection of future days does grow, even if it is not continually increasing, as new days are added to the collection as time progresses.

Regarding the third argument, it appears to rest on the assumption that the number of future days is finite. However, this assumption is not uncontroversial, as it presupposes a specific theory of time. You could argue that time is not like space, and thus, it is not clear that it makes sense to speak of the number of future days in the same way that one speaks of the number of objects in space.

My last comment didn’t seem to go through. I signed up for an actual account on this website instead of just using the ad hoc comment sign-in, so maybe that’s it. I apologize if I’m breaking any rules.

In the interest of joining the effort of helping people see what is seen by the people who make these types of arguments, I suggest the following clarification.

The critique of Craig’s argument I believe goes something like this:

With the definitions as they are given and following the thought processes as they are laid out in the blog post, I don’t disagree. However, I don’t think that this definition of actual infinity is all that Craig has in mind when we talk about an “actual” infinity. It seems to me that actual infinity here means something like “actually meeting the definition of infinity” whereas Craig means something about its physicality.

For example, if we consider a stick made out of wood, there is a sense in which we can say that this stick is an actually infinite collection of divisible stick lengths. This scenario matches the definition above, but why doesn’t the stick also pose a problem for Craig’s distinction?

A further distinction I think has to be made between infinities of continuous properties and infinites of discrete, physical units.

For example, if we had to map the stick’s divisible stick lengths onto the physical quanta of the stick, we would find that we cannot map them in a 1-1 correspondence. Our ‘collection’ only exists mathematically and conceptually.

In the case of the future, considered from our current point of view, in what sense does the future have any discrete reality at all? It only ever exists as something that has not yet arrived as a discrete, physical unit. It is as abstract a collection as the idea of an infinite collection of divisible stick lengths.

The past, however, is composed of events that have already arrived as discrete, physical units. It seems to me that this represents a clear asymmetry.

In the case of considering an infinite progression of future events, we are asked to consider an actual infinity of an abstract not-yet. In the case of considering an infinite regression of past events, we are asked to consider an actual infinity of discrete physical events.

So, it is relevant that George “never” actually reaches a point where he could say that he has counted an infinite number of objects, and that’s because we are being asked to believe that this is being accomplished right now. That is, every day that goes by is the end cap of an infinite series of discrete physical events. We can look back and say that an infinite series of discrete physical events has happened.

Thanks for the comment my dude!

You say: “In the case of the future, considered from our current point of view, in what sense does the future have any discrete reality at all?”

The past *also* doesn’t have any reality, under Craig’s presentism. There is no such thing as the past. There are no past events. They’re precisely nothing; they don’t ‘have reality’, any more than unicorns have reality.

You continue: “It only ever exists as something that has not yet arrived as a discrete, physical unit. It is as abstract a collection as the idea of an infinite collection of divisible stick lengths. The past, however, is composed of events that have already arrived as discrete, physical units. It seems to me that this represents a clear asymmetry.”

Sure, the past events ‘have already arrived’; but symmetrically, the future events *will* arrive, and so you haven’t pinpointed any asymmetry apart from the mere fact that one is past while the other is future, which isn’t in dispute.

You go on to claim that it is relevant that George never reaches a future point in time at which it is true that he has finished counting through all the natural numbers. That may be relevant to the argument from successive addition (i.e., the argument from the alleged impossibility of traversing an actual infinite); but it isn’t relevant to Craig’s *other* philosophical argument for the Kalam’s second premise — the argument from the impossibility of actual infinites — which is the only argument that the endless future criticism targets.

We need to keep in mind the dialectical context in which the endless future criticism arises. It arises in the context of Craig’s argument that actual infinites — that is, collections equinumerous with the natural numbers — are impossible. Once we grant that the collection of future years is equinumerous with the natural numbers — and it is — this collection constitutes an actual infinite and hence is impossible, per Craig’s argument. This stands regardless of whether there comes a time at which all the members of the collection have elapsed. What matters is not that they’ve all elapsed; what matters is that they’re equinumerous with the natural numbers. Once we grant that they’re equinumerous with the natural numbers, they thereby share with Hilbert’s Hotel (HH) the allegedly absurd properties attending actually infinite collections. And they have these properties regardless of whether there ever comes a time at which all of them have elapsed.

To see this, notice that one of the alleged absurdities of HH is that one can remove members from the collection of guests while the remaining collection of guests retains the same number of members after said removal. For Craig, this is absurd. And yet this is precisely what attends the actually infinite collection of future years — as years go by, the collection of future years loses members, and yet retains the same number of elements [namely, aleph-null]. Another of the alleged absurdities of HH is that one can ‘subtract’ identical quantities from the collection of guests and get divergent quantities of guests as a result. And yet this is precisely what attends the actually infinite collection of future years; we can easily imagine a counter who counts one number per day of the endless future [1, 2, 3, 4, 5, 6, …] and another counter who skips every other number [1, _, 3, _, 5, _, 7, …] and hence skips every other day of counting throughout the endless future. When we compare the number of future counts for the second counter to the first, we’ve removed infinitely many counting events from the first to the second. And yet the second counter still does infinitely many counting events throughout the endless future. Now consider a third counter who only counts 1, 2, 3, and then stops for the rest of eternity. Compared to the first counter, we’ve now removed infinitely many counting events [all those greater than 3], and yet we’re left with a finite number of counting events. So we’ve removed identical quantities [infinitely many counting events, in the case of the second and third counters] from an identical quantity [the first counter], but we’ve gotten divergent quantities as a result. For Craig, this is absurd. And yet it’s a direct consequence of the endless future. Finally, consider a third allegedly absurd feature of HH: the whole collection of guests is equinumerous with proper sub-collections thereof. For Craig, this is manifestly absurd. And yet this is precisely what attends the actually infinite collection of future years. The whole collection of future years is equinumerous with the proper sub-collection thereof consisting of every other future year.

So the allegedly absurd elements of HH equally attend the endless future, and nothing in this reasoning relies on there ever coming a time at which all of the members of the collection have elapsed.

If you’re curious, I explain this stuff in more detail in my Hilbert’s Hotel video here 🙂

https://www.youtube.com/watch?v=wt-rEeUIcR4

I appreciate you taking the time to respond to my comment. 🙂 I find value in your contributions. I have some thoughts on this response as well, but it may be the case that I’ll need to get clarification from Craig.

The past doesn’t have any reality because they are events that have come and have gone. The future doesn’t have any reality because they haven’t even come to even go.

These two notions of not having reality are differentiated by the way in which they intersect with the domain of the actually physical. Past events have at least been physical, whereas future events never are physical.

You say: “sure, the events ‘have already arrived’; but symmetrically, the future events

*will*arrive, and so you haven’t pinpointed any asymmetry apart from the mere fact that one is past while the other is future, which isn’t in dispute.”The asymmetry comes into play when we consider the property of physicalization. To make the claim that the past is an infinite regression is to make a claim about a series of discrete physical units. Future events are not discrete physical units even if one day they will be. Once they are, they are no longer future events, they are the past or the present.

Ten houses that were built and are gone are different than ten houses that were never built and never arrived. Ten of them leave a physical mark in the present, and ten aren’t physical at all.

In short, I think you’re right that whether or not they are physical or not is at best irrelevant when we consider them purely from the perspective of mathematics. An infinite future is not problematic in this sense. Neither is an infinite past.

I’ve taken the Calculus sequence along with many other math majors, so I am familiar with the idea of infinity, its mathematical self-consistency, and its validity.

But you also say that Craig’s argument “arises in the context of Craig’s argument that actual infinites — that is, collections equinumerous with the natural numbers — are impossible.”

Maybe this is a question I’ll have to find a way to ask Craig, because I don’t think that he would look at this definition of actual infinite and say such a thing is an impossibility in the context of, say, a stick with infinitely divisible stick lengths, or the variety of infinities used / assumed in the course of mathematics generally. He always pairs this impossibility with a clarification that he is referring to the physical implementation of a discrete series of events.

That is, it seems clear to me that each side is working with a different idea of “actual.” On one side, actual just means mathematically meeting the definition of infinite, which a consideration of the future as never-ending meets, but so do a lot of other things that are physically finite. This doesn’t have anything to do with how a thing physically exists, but for that reason I don’t think it’s what Craig means when he says that an infinite past cannot be actually infinite.

A collection? No problem. A collection of discrete, physical units? That is a horse of a different color.

I will watch that video, though. I have watched the first five or so minutes of the most recent video on this topic that was posted, and I will continue to watch that as well.

While it is true that Craig’s presentism entails that the past does not have any reality in the present moment, this is not the same as saying that the past does not exist at all. Craig believes that past events have occurred and have had real effects on the present moment. In contrast, the future has no such effects on the present moment and, therefore, does not have any discrete reality.

Furthermore, the criticism that future events will arrive symmetrically to past events is not relevant to Craig’s argument. Craig’s argument is not concerned with the temporal direction of events but with the idea of actual infinites. The future, as a collection of events, is equinumerous with the natural numbers and constitutes an actual infinite, according to Craig. This is what renders it impossible, not the fact that it is in the future.

Regarding your arguments that the endless future criticism targets Craig’s argument from the impossibility of actual infinites, you are correct. However, your claim that the argument from successive addition (i.e., the argument from the alleged impossibility of traversing an actual infinite) is not relevant to Craig’s philosophical argument for Kalam’s second premise is incorrect. The argument from successive addition is a key component of Craig’s overall argument for the impossibility of actual infinites.

Moreover, your analogy between the endless future and Hilbert’s Hotel is flawed. The absurdities of Hilbert’s Hotel do not simply arise from the fact that it is an actual infinite but from its peculiar properties, such as the ability to remove and add an infinite number of guests without changing the size of the hotel. The future, on the other hand, does not possess such properties and, therefore, cannot be equated with Hilbert’s Hotel.

In conclusion, while the endless future criticism raises important points, it ultimately fails to address the core of Craig’s philosophical argument for Kalam’s second premise.

I think for both sides it’s worth considering Aristotle’s reasons for thinking that mathematics doesn’t need actual infinities. Take your opening definition “Thus, an actual infinite is a collection whose members can be put in one-to-one correspondence with the natural numbers.”

Think hard about the word ‘can’ in that sentence. That sounds like a possibility. Natural numbers (esp for ancient greeks) are things like pebbles or strings intersected by something (like a finger placed on the string of a lyre at a definite interval.) Give me infinite pebbles, time and energy and, of course, I *could* place the number of future years in a 1 to 1 pairing. Now, interestingly Aristotle sides with Schmid on this one because, for Aristotle, God guarantees that there is infinite time, energy, and cyclical motions (the temporal equivalent of a pebble). However, he doesn’t think other kinds of infinite are actual: e.g. infinitely small or large magnitudes. You *could* walk to the edge of the cosmos and throw a spear, but you haven’t done that have you: you just thought about it. and so the infinity exists in your power of theoretical contemplation not in the actual world. Likewise, Achilles divide his journey across the stadium infinitely many times…(and stick a pebble there if you want) but he doesn’t actually do so.