Can something be infinite, yet also exist, as it were, ‘in reality’? Many people say that the answer to this question is ‘no’. However, the arguments in favour of this go over terrain that is very difficult to navigate without getting lost. Here I want to look at one very small part of the issue, and what seems problematic about it to me. As I said, this stuff gets very deep very quickly, and although I know the literature a bit, I am not a specialist in this area. There plenty of discussions of this I should probably read. But these are my thoughts at the moment. Hopefully it will help me become clearer about it, and might be helpful to people trying to understand this area a bit better too.
I am thinking about the argument in relation to the ‘Hilbert’s hotel’ style of defence for the second premise of the kalam cosmological argument. In particular, imagine that an apologist uses the infinite library analogy as follows:
Suppose there were a library with infinite books in it. If you withdraw a book from the library, then there is one fewer book in the library, yet there is also the same number of books in the library. There cannot both be fewer books and the same number of books, because that is absurd.
That is the type of claim that I am looking at here. I am not looking in particular about the other difficulties which could be brought out from the infinite object examples. There are other things one might say to motivate this part of the kalam, but here I am looking at this way of motivating it. The claim I am interested in is that the truth of there being fewer books and the same number of books is itself an absurdity.
- Equinumerous and fewer than.
There are two principles which need to be kept in sight. They involve connecting our intuitive ideas ‘equinumerous’ and ‘fewer than’ with mathematical counterparts.
We already have a fairly clear idea about the relationship between the two concepts when we use it in casual conversation. In particular, it seems quite clear that they are mutually exclusive:
i) If the number of A’s is equal to the number of B’s, then the number of A’s is not fewer than the number of B’s.
ii) If the number of A’s is fewer than the number of B’s, then the number of A’s is not equal to the number of B’s.
In ‘real life’ examples, we say that if the number of forks in my dinner set is the same as the number of knives, then I do not have fewer forks than knives, etc. The mathematical counterparts of these terms need to preserve this highly intuitive relationship; the result must be mutually exclusive too. We will consider two linking-principles, one which links the notion of ‘equinumerous’ to a mathematical idea, and one which links the notion of ‘fewer than’ to a different mathematical idea. They are chosen because they look like they express what the intuitive ideas are getting at, and because they preserve the mutual exclusivity relationship (at least, they do to begin with).
2. Hume and Euclid
The first of these is sometimes called Hume’s Principle (HP). It can be seen as a definition of the notion of ‘equinumerous’ (or ‘the same number of elements’, or of having the ‘same cardinality’ – all three meaning the same thing here). Being the same cardinality is linked to being able to be put in a one-to-one correspondence. Imagine that I could put every fork from my dinner-set with a unique knife, and have no knives left over. According to HP, this condition is what must hold for it to be true that I have the ‘same number’ of forks as knives. The idea with HP is that “The cardinality of A is equal to the cardinality of B” means that the elements of A can be put in a one-to-one correspondence with the elements of B.
‘Euclid’s maxim‘ (EM) effectively says that the whole is greater than a part (this principle is advocated by Euclid in the Elements). We can see this as a definition of the notion of ‘fewer-than’. Firstly, we need to be clear about what a ‘proper subset’ is:
A is a proper subset of B if and only if
- everything which is an element of A is also an element of B, and
- there is an element of B which is not an element of A.
So the set of knives is a proper subset of the set of ‘items of cutlery in my dinner-set’, because there are items of cutlery which are not knives (i.e. forks). There are fewer knives than there are items of cutlery. So this the idea with Euclid’s maxim is “There are ‘fewer’ A’s than B’s” means that A is a proper subset of B.
Here they are side by side:
Hume’s principle: A is equinumerous with B if and only if the elements of A can be placed in a one-to-one relation with the elements of B.
Euclid’s maxim: There are fewer A’s than B’s if and only if A is a proper subset of B.
3. The Problem
If the number of elements that can be in a set is finite, then these principles are mutually exclusive. So if A and B are equinumerous (according to HP), then neither is fewer than the other (according to EM), and if one is fewer than the other, then they are not equinumerous. This is because, if A and B are finite, then there being something in B that is not in A entails that one could not put their respective elements in a one-to-one relation. If I am missing a knife, then I cannot place each knife with a unique fork, without having a fork left over. So far so good.
The problems come in if A and B are allowed to be sets that have infinitely many elements. When we make this move, these two intuitive principles cannot both be correct. Let A be the set of all natural numbers, [0, 1, 2, 3 … n …), and B be the set of even natural numbers, [0, 2, 4, 6 … n …). The elements of A and B can be placed in a one-to-one correspondence with one another, as Cantor showed. So by Hume’s principle, they are equinumerous. Yet, it is also clear that every element of B is an element of A, while there are elements of A that are not elements of B (i.e. the odd numbers). This means that B is a proper subset of A. So by Euclid’s maxim, B has fewer elements than A.
The problem is that A and B have the same number of elements, but B has fewer elements than A. As we saw, the intuitive relationship between being the same number and being fewer is that they are mutually exclusive. So it should never happen that A and B are both equinumerous and that one is fewer than the other. Clearly, something has to give here if we are to avoid an inconsistent result.
4. What is going on?
The issue here is that we have a pre-theoretical idea of the terms ‘equinumerous’ and ‘fewer than’, and we have set-theoretical expressions which looked like they gave the meaning of the intuitive notions. However, our intuitions about what those terms mean differ from how their mathematical counterparts operate in certain circumstances.
So to avoid the problem, we have to reject one of three things:
- The idea that equinumerous means being able to be placed in a one-to-one correspondence (i.e. HP)
- The idea that fewer-than means being a proper subset (i.e. EM)
- The idea that being ‘equinumerous-with’ is mutually exclusive with being ‘fewer-than’.
If we try to keep all three of these things, we run into the problems that give rise to the apologist’s charge of ‘absurdity’.
5. Rejecting 1
One way of proceeding is to reject HP. This means rejecting the claim that when we say that the number of A’s is the same as the number of B’s this means that the A’s can be placed in a one-to-one correspondence with the B’s. The main problem with this is that it is unclear what else being ‘equinumerous’ could mean. Possibly, it could mean something like if they were both counted, then the final number reached would be the same in each case. It is not clear whether this is actually any different however. Imagine that I count my knives by picking each one up and saying a cardinal number out loud (like the Count from Sesame Street), and then placing them off to one side in a line according to the number they received. So I put the first one down, then I place the second one next to that, and the third one next to the second one, etc. When I come to count my forks I could do exactly the same thing. If I arrive at the same number when I have finished counting each one, this just means that the two lines of cutlery would be lined up one-to-one. So this doesn’t even seem to be a different result to HP. And what else could ‘equinumerous’ mean?
The real action is about which one to reject out of 2 or 3.
6. The case for rejecting 2
One way to reject 2, but to keep 3, would be to modify the claim made in EM. At the moment, EM says that there are fewer A’s than B’s iff A is a proper subset of B. We could add another condition as follows:
Revised-EM) There are fewer A’s than B’s iff
- A is a proper subset of B, and
- A and B are not equinumerous
The second condition isn’t needed in the case of merely finite sets, because no finite set A can be both a proper subset of B and equinumerous with B. Thus, the original EM and the revised-EM are identical with respect to finite sets. When we move to the case where sets can be infinite, then the second condition kicks in. The set of the even natural numbers is a proper subset of the natural numbers (so the first condition is satisfied). But the set of even natural numbers is equinumerous with the set of natural numbers (in that they can be placed in a one-to-one correspondence). Because this second condition is not satisfied, this means that it is false that there are ‘fewer’ even natural numbers than natural numbers. And this means that, according to revised-EM, there is no case (finite or infinite) where A is both equinumerous-with and ‘fewer-than’ B. And thus we have resolved our problem.
According to this strategy, there is nothing wrong with equinumerous meaning being able to be put in a one-to-one relation, and there is nothing wrong with the intuitive idea that equinumerous and fewer-than are exclusive. All that is rejected is the assumption that all there is to the notion of ‘fewer-than’ is being a proper subset. In addition to this, we also need to rule out being equinumerous. Only with both in place do we have a proper mathematical equivalent of ‘fewer-than’.
7. The case for rejecting 3
On the other hand, we could proceed by rejecting 3, the mutual exclusivity of equinumerous and fewer-than. On this view, infinite sets show us clear examples of when the A’s are equinumerous with the B’s, even though the A’s are also fewer-than the B’s. One might argue that our intuitions about the relationships between these terms is based on our experience of finite things, and we mistook a property of finite things to be a logical relationship between two terms. The mutual exclusivity of equinumerous and fewer-than is not a logical truth, but is actually a contingent truth, which applies only to those cases where the sets are finite. According to this view, we should revise our notions in light of this mathematical insight.
So take some case involving infinity, such as the infinite library. The number of books left in the library after I withdraw one book is ‘fewer’ (i.e. according to the original EM) than the number of books before the withdrawal, even though there is also the same number as before the withdrawal. There is only a problem with this if you insist on the mutual exclusivity between ‘same number as’ and ‘fewer-than’. If we let go of that presupposition, and let the mathematics guide our understanding, we see that the two notions are only mutually exclusive for certain cases and not others.
This sort of revision in how we use terms guided by scientific insight is not that strange. Imagine that at some point in history we discovered androgynous frogs. Prior to that we would have said that the terms ‘male’ and ‘female’ were mutually exclusive when it came to classifying frogs; if a frog is male, it is not also female and vice versa. But after the discovery we have a choice about how to proceed. We do not, I take it, say that these are not frogs, merely because it is true that no frog is both male and female! Rather, we say that, despite what we have previously thought, ‘male’ and ‘female’ are not mutually exclusive for all frogs. We revise our understanding of ‘male’ and ‘female’, being led by the discovery.
This is what it is like in our case too, if we reject 3. We originally thought that no sets could be fewer-than and equinumerous, but this was only the case with the finite sets we had considered. Once we look at these other cases, we find out that some sets are both fewer-than and equinumerous. Once we accept this, and drop the requirement that they are always mutually exclusive, we have avoided our issue from before. Saying that the library has both the same number and fewer books is like saying that this frog is both male and female. Sometimes that is what it is like.
So we have two strategies. The disagreement is over the following. Imagine A is an infinite proper subset of B, such as A being the even natural numbers and B being all the natural numbers. A is equinumerous with B. But are there ‘fewer’ A’s than B’s? The first strategy says:
No, there are not fewer A’s than B’s (because they are equinumerous)
The second strategy says:
Yes, there are fewer A’s than B’s (because one is a proper subset of the other)
It is fairly clear at this stage that if one wanted to use the Hilbert’s hotel argument as a way of bringing out an absurdity, then option 2 causes a big difficulty. This is because it denies that there is ever a case where any two sets can be equinumerous and fewer-than. In particular, the infinite case is protected from this happening by the second condition in revised-EM. In these cases, the equinumerous nature of the two sets cancels out either being fewer-than the other. The very thing the apologist wanted to point to and say ‘Look at this! It’s absurd!’ is forbidden on this view.
Indeed, the third option also causes grave issues for the apologist too. On this view we have revised our notion of ‘fewer-than’ in such a way that it is no longer mutually exclusive with ‘equinumerous’. It would be like after the discovery of androgynous frogs; if I say to you ‘This frog is male’, you could reply ‘Yes, but is it also female?’ This reply wouldn’t be ‘absurd’ at all, because these terms are no longer thought to be mutually exclusive. The same thing would apply in our case too. On this view, there being both the same number and fewer books in the library after I withdraw one is not an example of two mutually exclusive things being true at the same time. Therefore it is not absurd on this view either.
So the issue I have looked at in this post is only a very small issue in the wider context of defending the kalam. It isn’t even the only issue that is brought up in relation to the Hilbert’s hotel style of defence, or even arguably the most serious. However, it is there, and people often talk as if this issue on it’s own causes problems. People often talk about the absurdity of there being both the same number of books and fewer books after the withdrawal in this sort of setting, even if they also develop additional worries.
I think there are broadly two strategies that one can adopt in response to this line of attack. The first would be to insist that there are really no situations where there are both the same number and fewer books, and provide a precise explanation of ‘fewer-than’ according to the revised-EM above. This clearly avoids the issue. Secondly, one could embrace the presence of what seemed like two mutually exclusive terms, but explain how the mathematics shows us that the two terms are not mutually exclusive for all cases.
I find each of these approaches to be independently quite plausible, and this largely discharges the force of the attack.