More on the actual / potential infinite

0. Introduction

One of the premises of the Kalam Cosmological Argument (KCA) is that the universe began to exist. There are two types of defence for this premise; scientific and philosophical. In the latter category, there is one argument in particular that I want to focus on, which Craig calls the ‘argument from the impossibility of an actual infinite’.

The argument runs like this:

  1. An actual infinite cannot exist
  2. An infinite temporal regress of events is an actual infinite
  3. Therefore, an infinite temporal regress of events cannot exist

Craig holds that the past had a beginning, but also that the future has no end (presumably due to his beliefs about the afterlife). This invites the following objection, which has been made in the literature by Wes Morriston (here). We seem to be able to formulate a symmetrical argument which should conclude that the future has an end point:

  1. An actual infinite cannot exist
  2. An infinite temporal progress of events is an actual infinite
  3. Therefore, an infinite temporal progress of events cannot exist

(The term ‘progress’ is artificial used in this context, but it is clearly intended as the temporal mirror of the term ‘regress’)

The first argument says that the past must have a beginning, otherwise it would constitute an actual infinity. The second argument, the counter-argument, says the future must have an end, otherwise it would constitute an actual infinity.

Morriston has a thought experiment to illustrate his point. He asks us to imagine two angels, Gabriel and Uriel, who take turns saying praises to God forever. He makes the following remarks:

“It’s true, of course, that Gabriel and Uriel will never complete the series of praises. They will never arrive at a time at which they have said all of them. Indeed, they will never arrive at a time at which they have said infinitely many praises. At every stage in the future series of events as I am imagining it, they will have said only finitely many. But that makes not a particle of difference to the point I am about to make. If you ask, “How many distinct praises will be said?” the only sensible answer is, infinitely many.” (Morriston, Beginningless Past, Endless Future, and the Actual Infinite, p. 446)

To counter this, Craig argues that the endless future is best considered a merely potential infinity, (in contrast to the beginningless past, which is best considered as an actual infinity). As Craig says in his reply to Morriston, Taking Tense Seriously:

“So with respect to Morriston’s illustration of two angels who begin to praise God forever, an A-theorist will concur whole-heartedly with his statement, “If you ask, ‘How many praises will be said?’ the only sensible answer is, infinitely many”— that is to say, potentially infinitely many. If this answer is allowed the A-theorist, then Morriston’s allegedly parallel arguments collapse.”

Effectively, Craig is denying the second premise of our counter-argument. He is saying that an infinite temporal progress of events is not an actual infinity – it is merely a potential infinity.

In what follows I want to look at three types of response to this. This post will constitute the first part, and in subsequent posts I will address the second and third points.

Firstly, I will spell out an intuition that many people have, according to which the existence of a potential infinity entails a corresponding actual infinity. We will call this ‘Cantor’s Intuition’ for reasons we will get into below. If Cantor’s Intuition was correct, then Craig’s response would be defused. For then we could “concur whole-heartedly with his statement” that the future is potentially infinite, and insist that it is also actually infinite. According to this line of thinking, the potential and actual infinite are not mutually exclusive.

Secondly, I want to look at a different strategy. Perhaps the future is not actually infinite, and the second premise of the counter-argument is false. But the thought is that maybe this leaves open the door to denying the second premise of the original argument. That is, if the infinite progress of events is a merely potential infinity, maybe the infinite regress of events is a merely potential infinity as well. Craig is very dismissive of this view, but I think it is worth exploring.

Lastly, I want to look at why Craig thinks there is an asymmetry here at all. Here it seems that considerations about the philosophy of mathematics are completely irrelevant, and all that is doing the heavy lifting is considerations about the philosophy of time. Of course, all but the most fanatical would concede that time is in some sense asymmetric. Yet, this can be cashed out in lots of different ways. Do any of those ways of understanding the asymmetry do the work that Craig needs?

  1. Potential infinite implies actual infinite

The mathematical study of the infinite was revolutionised in the 19th century by the work of various mathematicians, but the primary figure is clearly Georg Cantor. He was the first to work out the mathematics of the infinite, and in particular gave a formal treatment of the actual infinite. He changed his mind quite a lot, but at one time in particular he held a view that I want to bring up here. In 1886, he wrote a letter to the mathematician Richard Dedekind, in which he made the following comments:

“… since there can be no doubt that we cannot do without the variable magnitudes in the sense of the potential infinite, then the necessity of the actual infinite can be proved as follows: In order for such variable magnitudes to be capable of evaluation in a mathematical investigation, their “range” of evaluation must be precisely known by means of a prior definition. But this “range” cannot itself be in turn something variable, for otherwise every fixed support for the investigation would give way; hence this “range” is a definite actually infinite set of values. Thus, every potential infinite, if it is to be employable in mathematics, presupposes an actual infinity.” (Quoted in ‘The Potential Infinite‘, by W D Hart, 1979)

Cantor’s Intuition seems to be that the following inference is valid:

x is a potential infinite; therefore, x is an actual infinite.

If Cantor’s Intuition is right, and the above inference is valid, then Craig’s argument does not work. The reason is that Craig is using the terms potential infinity and actual infinity as if they were mutually exclusive; that something can be one or other but not both. After all, he says that the endless future is potentially infinite as a rebuttal to Morriston’s claim that there are infinite future prayers. Clearly, Craig thinks this rebuttal rules out there being actually infinitely many future prayers.

But if Cantor is right, then something being potentially infinite means that it is also actually infinite. If so, then when Craig says that the endless future is potentially infinite, this would entail that it is also actually infinite. And that would completely undercut his reply to Morriston.

But is Cantor right?

2. A potential infinite that is not an actual infinite

Well, in some sense it seems that he was wrong. Not every potential infinity presupposes an actual infinity. Consider the hierarchy of sets. So, start with the empty set (the set with zero elements):

Then there is the set with one element; namely, it has the empty set as it’s sole element:

{∅}

Then there is the set that contains two elements: the empty set, and the the set which contains the empty set:

{∅, {∅}}

The next level in the hierarchy contains the previous levels as distinct elements; levell 0 contains nothing, level 1 contains level 0, level 2 contains level 0 and level 1, etc. Clearly, we can go on elaborating this hierarchy forever, just constructing more sets in this way. But does this constitute a completed totality – an actual infinity?

There are good reasons for thinking not. Such a proposal seems to require that there is a ‘set of all sets’, and that seems incoherent. The reasoning is as follows. Suppose there was a set, V, which was the set of all sets. Well, why can’t we make another set, which has all the sets which are elements of V, as well as V, as it’s elements? Such a thing seems to be a set, and seems to employ just the process we have used at the lower levels. Yet it contains V, which we just postulated was the set of all sets. And that would mean that V is not the totality of all sets after all, but merely one more level of the hierarchy.

Such considerations seem to suggest that there cannot be a set of all sets, conceived of in this way. And if that is right, then the hierarchy of sets is potentially infinite, in that each set is finite but part of a never-ending hierarchy, where the notion of the completed totality is incoherent. Thus, along this way of thinking, we have an example of a potential infinite which is not an actual infinite. Such is the view of many people, including set theorists such as Ernst Zermelo, Kurt Gödel, and philosophers of mathematics such as Hilary Putnam, Charles Parsons, Geoffrey Hellman, and Oystein Linnebo.

And if this way of thinking is right, then Cantor was wrong here. Not every potential infinity implies an actual infinity.

3. A potential infinity that is also an actual infinity

Yet, things are not quite so straightforward. Although not every instance of a potential infinity presupposes an actual infinity; still, some might do. The hierarchy of sets is a particularly striking example where the idea of the completed infinity seems incoherent (for the reasons given above). However, the same sorts of considerations are not present in other cases.

For example, take the natural numbers. One can easily, and quite without contradiction, talk about the set of ‘all natural numbers’. This notion does not fall prey to the same worries as the set of all sets. Part of the achievement of Cantor was to elaborate the mathematical treatment of totalities such as the natural numbers. It is true that one could imagine counting forever, and such a process would increase without limit, always remaining finite and never being completed. Thus, it would be a potential infinity.

However, we can say things like “You will never have counted all the natural numbers” and when we use the phrase ‘all the natural numbers’ so we refer to a coherent concept. Even if we cannot reach the totality by counting, the concept of the totality itself does not seem incoherent in the same way as it does for the hierarchy of sets.

So in the case of the natural numbers, we have a potential infinity (instantiated by you trying to count them all), but we also have a completed infinity, which is the totality of numbers you are counting. And this is Cantor’s point. You can have a ‘variable magnitude’, which is the number you have counted (which is increasing over time), and there is the ‘range’ of numbers you are counting off, which does not increase and is an actual infinity. Thus, it seems like a potential infinity which presupposes an actual infinity.

Some people do disagree with this, of course. But such people are not merely saying that the concept of the actual infinity cannot be applied in the ‘real world’, as opposed to the mathematical world. Rather, such opposition requires disagreeing with Cantor that the actual infinity is a legitimate concept even in the mathematical realm. Carl Friedrich Gauss, for instance, strongly objected to the actual infinite even in mathematics. Such a position is called ‘finitism‘.

Craig, on the other hand, seems to have no principled objection to Cantorian treatments of the actual infinite in mathematics; he does not seem to be a finitist. If so, he should accept that sometimes there are potential infinities that are also actual infinities, such as the natural numbers.

4. Conclusion

Where does this leave us? I think we can say two things. Firstly, the following inference is invalid:

If x is a potential infinity, then x is an actual infinity.

It is invalid because the hierarchy of sets seems to be a plausible counterexample. However, unless one wants to take a very stern Gaussian position and banish the actual infinite even from mathematics, one must also concede that the following inference is also invalid:

If x is a potential infinity, then x is not an actual infinity

This seems invalid because the coherence of the totality of the natural numbers seems to be a counterexample.

This means that one cannot say that the angels prayers constitute an actual infinity merely because they constitute a potential infinity; but also one cannot say that they do not constitute an actual infinity merely because they constitute a potential infinity. Both sides can agree that they constitute a potential infinity, and this leaves open the question about whether they also constitute an actual infinity. In effect, the observation that they constitute a potential infinity is besides the point. The salient issue is about whether they constitute an actual infinity, and that is logically independent (assuming both of the above inferences are indeed invalid).

I think the lesson from this is that some potential infinities are also actual infinities, and some are not. The question becomes: which type is the future? The case we saw where something was potentially infinite but not actually infinite involved an incoherence involved in the notion of the totality. Is such a consideration present when it comes to the notion of the future?

One thing that seems plausibly problematic is the notion of the last time. One might think that the very notion of ‘a time’ implies that it has a past and a future. Such seemed to be Aristotle’s view:

“Now since time cannot exist and is unthinkable apart from the moment, and the moment a kind of middle-point, uniting as it does in itself both a beginning and an end, a beginning of future time and an end of past time, it follows that there must always be time: for the extremity of the last period of time that we take must be found in some moment, since time contains no point of contact for us except the moment. Therefore, since the moment is both a beginning and an end, there must always be time on both sides of it.” (Physics, book 6, part 1)

But such considerations shouldn’t sway us here. After all, the notion of number is similar in this respect. Just as we might think that for each moment of time there must be both past and future on either side of it, so too for each number there must be both higher and lower numbers on either side of it. The point is that we can conceive of the totality of natural numbers without thinking of there being a highest natural number. So, by analogy, even if there is no final time, this does itself stop us from conceiving of the totality of all future time.

We would be able to say that the future is potential and not actually infinite if there were some incoherence involved in thinking of the totality of future time, like there was with the totality of the hierarchy of sets. Yet, the cases seem dissimilar here. After all, what was causing the problem with the case of the sets was that the totality was itself a set. This meant that it could be fed into the iterative process that generated each preceding level in the hierarchy, generating a new level above it.

But such a move is not applicable to the notion of time. After all, the totality of time is not itself a time. Therefore, we need not suppose that the totality of time is itself followed by another time. If we did, then the case would be analogous to the set example. But it seems clearly to be distinct.

This does not establish that the case of time definitely is one that is both potentially infinite and actually infinite, but it does seem to show that if there is a reason it is not directly analogous to the hierarchy of sets example. Maybe there is an argument, but what is it?

My thought is that the time example is more like the natural numbers than the sets. Talk of the totality seems coherent. Thus, it seems entirely possible, at least conceptually, that the future is both a potential infinity and an actual infinity. And if that is right, then Craig’s reply is kind of impotent. Yes, potentially infinitely many praises will be said. But also, there is an actual infinity of praises yet to be said. The former point does not itself rule out, or in, the latter. Clearly, more needs to be said (though, hopefully not infinitely more).

8 thoughts on “More on the actual / potential infinite”

  1. A simpler objection to Aristotle’s objection might be to note that we are already supposing that there is a first moment. Thus, the whole scenario is premised on the assumption that no such objection can be valid; if it were, we’d have to deal with infinite REgressions.

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  2. Hi Alex,

    I am a little bit confused about your example involving sets. To me, it seems like the construction you outline is actually parallel to the construction of the natural numbers.

    Let me define an operator P which acts on sets in the following way: for any set S,

    P(S) = S ∪ {S}

    where ∪ means “union”. Then in your text, you basically begin with the empty set, and repeatedly apply the operator P. It seems to me that the set you create in this way, by “elaborating this hierarchy forever”, would more accurately be described as follows:

    V is a set, each one of whose elements is a set obtained by applying the operator P to the empty set a finite number of times.

    By saying that we are “elaborating this hierarchy forever”, we are basically saying that V contains all the sets that you can make by repeatedly applying the operator P. But you can only ever apply the operator P a finite number of times in each case. So in some ways this still seems like a “potential infinite”, but on the other hand V clearly contains an infinite number of elements. Thus V incorporates both an “actual” and a “potential” infinity, in some sense.

    We can also see that this avoids the problem of the “set of all sets”. Consider the question: does V (as I defined it above) contain itself? Well, if it contained itself, this would mean that we could construct the set V by some finite number of applications, say N applications, of P to the empty set. But clearly this is not true: V also contains the set obtained by applying P (N+1) times. In this way, this construction using sets really mirrors the construction of the natural numbers, and the set V is similar to the set of all natural numbers, each one of which can be reached by counting upwards some finite number of steps.

    You might wonder what happens if we consider the set V ∪ {V}. This would be like taking the set of natural numbers and then adding an additional element ∞. There is nothing wrong with this on the level of sets, but we cannot expect this additional element to follow the same rules as the other elements of the set – for example, it is not clear how to “count up or down” from ∞, whereas this is built into the construction of the natural numbers, which are the other elements of this set.

    My own thoughts on this actual/potential infinity question are the following: as a mathematician, I was rather surprised to discover this concept, which I have never heard discussed by mathematicians. But the more I look into it, the more I feel like every time someone tries to back up a claim that there is something wrong with an “actual infinity”, their example of an “actual infinity” is really just an incoherent statement which tries to treat infinity as though it were a number.

    Take the following argument, variations of which I have heard or read a few times. “The past cannot be infinite, because if it were, we could never have arrived at the present.” What does this mean? Suppose the past were infinite. To me, this means something like the following: for every natural number N, the world existed N days ago. Now, the argument that we would never reach the present goes something like this: start with N = ∞ and count backwards, and you will never reach 0, that is, you will never reach the present. But N cannot equal ∞, N is supposed to be a natural number! In other words, we are only supposing that the universe existed a finite number of days ago, but where this finite number can be anything we like – that is what it means to exist forever in the past! It certainly does not mean that there is some natural number ∞, and that the universe existed ∞ days ago, which is just incoherent.

    I think that the reason for the injunction against “actual infinities” arises from this kind of confusion. People realise that there is something wrong with treating ∞ as a number, because it can lead to all sorts of contradictions. But people mistakenly think that these contradictions arise from the “physicality” or “actuality” of the situation they are imagining – for example, an infinite past would be an “actual” infinity because the past has some kind of ontological status which differs from, for example, the natural numbers. On the other hand, if you pretend that ∞ is a number and try to use it mathematically, you will run into contradictions just as fast. In the latter case, everyone can agree on what went wrong: you illegitimately used ∞ as if it were a member of a set (say the natural numbers) of which it is not a member.

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    1. Hi Joe,

      You say:

      ///It seems to me that the set you create in this way, by “elaborating this hierarchy forever”, would more accurately be described as follows:

      V is a set, each one of whose elements is a set obtained by applying the operator P to the empty set a finite number of times.///

      V is a set that has that property. But that isn’t the definition of V. It’s only a necessary condition, but it isn’t sufficient. Consider level ‘1’ (or the second level), which is:

      {∅}

      It satisfies that definition, because it is ‘a set each of of whose elements is a set obtained by applying the operator P to the empty set a finite number of times’ (namely one time). But, surely, {∅} ≠ V.

      Each element of V is obtained by finite iterations of P, but, crucially, V is the result of applying the P operator to the empty set *as many times as possible*. By ‘possible’, we mean mathematically possible, not physically possible. Obviously, I would run out of ink, or grow bored, or die, or whatever eventually if I tried to write it down. But nothing about the mathematics itself ever imposes a limit. And that is the relevant sens of possible here – it is mathematically possible.

      Here is how Oystein Linnebo, a prominent philosopher of mathematics, puts it:

      “But how far does this hierarchy extend? … We are often told that the hierarchy extends as far as possible. Vague though this may be, it is hard to see how a more definite answer could be provided. For given any attempt to pinpoint the extent of the hierarchy in a more definite way, it seems possible for the hierarchy to extend even further by allowing any objects from the proposed characterization to form a set. And since the hierarchy is supposed to extend as far as possible, this means that the proposed characterization cannot have been correct after all. So there seems to be something inherently potential about the set theoretic hierarchy. Given any attempt at a definite characterization, it turns out to be possible for the hierarchy to extend even further.” (The Potential Hierarchy of Sets, Linnebo, 2013, p 205).

      So you say:

      ///We can also see that this avoids the problem of the “set of all sets”.///

      But it only does that because it isn’t a definition of V. It’s a condition that applies to each level of V, but doesn’t characterise V. It’s true that no level contains itself (that’s one of the reasons set theorists originally came up with it). But that observation doesn’t avoid the problem I brought up (which Linnebo also explains above).

      I agree with you to a large extent about what is happening when people talk about why the past cannot be infinite. You can’t ‘start’ counting backwards from ∞. The usual example given (which comes from Wittgenstein I think) is about finding someone who has been counting forever, who is just now saying “-3, -2, -1, 0. There, I finished!” Now, this is obviously a really weird idea. But if we say that he has been counting forever (in the sense that for any amount of time in to the past we pick, he was already counting at that point) it’s hard to see any contradiction. The best Craig brings up is that there would be a flouting of the principle of sufficient reason. Why had he not finished 5 mins earlier, or 2 years later? True, that seems to be impossible to answer. But picture someone else who finishes their countdown 5 mins earlier. We can say that this guy was always ahead of the first guy, by 5 mins. That is why he finished earlier. Again, no contradiction as such. If we picture one of Morriston’s angels beginning his count up the integers now, he would continue forever, and if his buddy waited five mins and then began his count, this would explain why his count is always behind. But why did they start at these points and not some others? That doesn’t seem mysterious. That is just when it started. So why is the reverse so mysterious? I’m not sure. It’s a hard area.

      Not sure if that helps or not! Let me know.

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      1. hello alex malpass, can I email you a critique of premise one of the KCA?

        I think that there is a very big error in Craig’s argument and I would be grateful if you could critique my argument?

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  3. Hi Alex,

    Thanks for the reply. I’m afraid that I was very clumsy when I tried to define V: what I should have said is the following:

    V is a set containing ALL of the following elements: the sets that can be obtained by applying the operator P to the empty set a finite number of times.

    This was the set that I had in mind. In this case it is easy to see that V contains an infinite number of elements. But it is still the case that V is a set, and that it does not contain itself. This is quite easy to see: suppose that V itself is an element of V. Then that would mean that the set V could be obtained by applying the operator P a finite number of times, say N times. But V also contains the set that is formed by applying P (N+1) times, since (N+1) is also finite, which gives a contradiction. So V is not an element of V.

    I think the point is that you can still enlarge this set by a similar operation. In other words, you can consider the set V ∪ {V}. That is, you can apply the operator P to the set V, which gives you a new, bigger set. And then you can repeat the whole process “at a higher level”: you could define the following set:

    V_2 is the set containing all of the following elements: the sets that can be obtained by applying the operator P to the set V a finite number of times.

    and then you could do that again, to form a set V_3, and so on. You could even do the following: define

    V_infty is the union of all the sets V_n, for all finite numbers n.

    But then you can still apply the operator P to find a bigger set!

    The real definition of V is supposed to include “all” these applications of P, but it’s hard to even say what this means – as soon as you write down some kind of useable definition, you find that you can enlarge this set by another application of P. This construction reminds me very much of Gödel’s incompleteness theorem.

    I agree with your thoughts on the infinite past. With regards to the person counting down forever and reaching 0 now: I feel like the “reason” that they reach 0 now is because that is how they are defined in this thought experiment. I think that Craig would like to “define” them only as someone who has been counting down forever at some specified rate – but this doesn’t give a sufficient definition. What number would they reach yesterday? Or how about 100 years ago? If we only define them as “a person who has been counting down forever” then we cannot answer these questions. This is perhaps slightly counterintuitive – if we suppose that they have been counting down from some finite number at a specified rate, starting some finite time in the past, then this is sufficient to answer those kinds of questions. But when we change to the infinite past then we need some extra information to be able to answer these questions – for example, if we know what number they reached 100 years ago, then that would be sufficient to define this counting person.

    I’m not sure whether this constitutes a violation of the principle of sufficient reason. If some object in a thought experiment is not fully defined, then we shouldn’t expect to be able to give an explanation for all of their actions. But the moment we do fully define them – for example, by saying that they reach 0 in the count now – it no longer seems mysterious that they reach 0 now.

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  4. Hey Alex,
    Sorry if I bothered you with my earlier comments. I plan to post an upgraded version of the original comment I mentioned above.
    I doubt it’s because it’s an old post as I saw others do the same, right?
    Anyways, I think you are an interesting guy and I look forward to reading more of your stuff.
    Thanks,
    AnonDoc
    P.S. Is there a limit to the number of links per comment? Are we allowed to post pics, vids or tweets?

    Like

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