Sunday, 23 May 2010

Infinity

I thought that it might be useful to gather together a few remarks about how infinity is considered in modern mathematics.

Underlying the mathematical treatment of infinity, there are a couple of fundamental observations about the nature of number.

The first is that of “1 to 1 correspondence”. When we say that there are three things sitting on the table, there is some notion that we can count them; I may even count them on my fingers. This is an example of putting the objects on the table into 1 to 1 correspondence with three of my fingers. Similarly, there is a very fundamental notion that any three objects may be put into 1 to 1 correspondence with any other three objects.

The second fundamental observation concerns the idea of “succession”. We think of the number three as being the number after the number after one. Indeed, as we count, we count off the numbers in succession: after any whole number is its successor.

Corresponding to these two fundamental notions, we get the idea of “cardinal number” and “ordinal number” respectively. Informally, a cardinal number measures how big a particular set is (in the sense of measuring how many elements there are in the set by putting its elements into 1 to 1 correspondence with a set of known size), and an ordinal number measures a set by putting its elements in order and counting them off in succession.

You’ll have noticed that when you count things on your fingers, you often use both concepts at the same time! You put the elements of the set of things into 1 to 1 correspondence with your fingers and you also order the set of things using the implied order on your set of fingers. However, notice that in putting objects in 1 to 1 correspondence with your fingers, there’s no need to use your fingers in any particular order (so the notion of cardinality and ordinality are at least notionally distinct). Notwithstanding this, there is a deep relationship between ordinal and cardinal numbers and this relationship remains true for non finite sets, but I’ll have to leave discussion of this fact to the textbooks.

Once we’ve got these ideas of cardinal numbers and ordinal numbers fixed, we can now start thinking about sets that are bigger than everyday finite sets. For example, consider the set that has all the natural numbers as its members: A={0,1,2,3,...}. Clearly this set is bigger than any finite set, but how big is it? Are there any sets bigger than any finite set but smaller than this set A?

To make sense of these questions, I’ll have to introduce a little bit of terminology. Let f: A --> B represent a correspondence (like the 1 to 1 correspondences we’ve been talking about above). So f takes an element a of the set A and pairs it with an element b of the set B (which we write f(a)=b). We say that f is a “surjection” when every element of B is paired with an element of A. We say that f is an “injection” when any element of B that is paired with an element of A is paired with a unique element of A. A 1 to 1 correspondence (also called a “bijection”) is then both a surjection (every element is paired) and an injection (but uniquely).

With these ideas defined I can now say that two sets are the “same size” if there is a bijection between them. I can also say that set B is “bigger” than set A if there is an injection from A to B, but no injection from B to A. Intuitively speaking, there are more elements in the set B than there are in the set A. You start off pairing the elements of set A with elements of set B, but you run out of elements from set A before you finish the elements of set B.

Having made sense of the notion of “bigger” and “smaller”, back to our set A={0,1,2,3,...}. In answer to our two questions, it can be shown that there is no non-finite set smaller than it and hence it is the smallest non-finite set. We will assign a symbol to its size: aleph-zero, the smallest “transfinite cardinal number”.

We’ve only got one of them so far, but we can already start to think what it means to do arithmetic on transfinite numbers. If we think about finite sets, we observe that if we have two sets of distinct objects then the number of elements in the two sets put together (in their “union”) is equal to the sum of the number of elements in each of the sets taken separately. So, if we have four things on the table and three things on the chair we have seven things altogether. We can write this as card(A) + card(B) = card(A union B). We can now simply take this as the definition of cardinal addition.

So, for example, what is 3 + aleph-zero? To answer this question, we take a set with three elements in it, such as {potato, chair, pen} and form the union of it with a set with aleph-zero elements in it, such as A={0, 1, 2, ...}, to form {potato, chair, pen, 0, 1, 2, ...} = C. I hope that it is clear to you that this set C can be put into 1 to 1 correspondence with A. Hence it is the same size as A, therefore 3 + aleph-zero = aleph-zero.

We’ve claimed that there are no transfinite cardinals smaller than aleph-zero; are there any bigger? Yes, in fact there are infinitely many larger transfinite cardinals. The easiest way of generating some of them is through the process of exponentiation. For finite sets, it can be shown that the set of all subsets of a set with n elements in it has size 2 tot he power n. So, we extend this to a definition: if a set X has cardinality k, we define 2 to the power k to be the cardinal of the set of all subsets of X. Now, for finite sets, it’s clear that 2 to the power k is bigger than k. This remains true for any set.

(Proof: suppose g:A-->set of all subsets of A, is a bijection. Form the set B={x in A such that x is not contained in g(x)}. Now clearly B is a subset of A, so there must be an element y of A such that g(y) = B. Now think about whether y is an element of B: if it is, it isn’t; if it isn’t, it is! Contradiction, so such a bijection cannot exist).

So, 2 raised to the power of aleph-zero is bigger than aleph-zero. We’ll call it aleph-one. 2 raised to the power aleph-one is bigger than aleph-one. We’ll call it aleph-two. And so on! In fact, aleph-one is the cardinality of the continuum (that is, the set of all real numbers, or equivalently of the set of all points on a geometrical line).

(Optional mind blowing fact: we can ask whether there are any cardinal numbers bigger than aleph-zero but smaller than aleph-one (this is the “continuum hypothesis”). In fact, whether there is or is not such a cardinal number cannot be proved from the normal axioms of set theory! One can define two different types of set theory: one in which you assume there is such a cardinal as an axiom and one where you assume the opposite.)

You can also define transfinite ordinal numbers, but I thought you would have probably had enough by now!

Now, much of this theory of the infinite was worked out during the latter part of the 19th century but it’s surprising to learn how close the medievals came to working out this theory before the Black Death and the Renaissance put an end to this sort of intellectual inquiry. So Saint Thomas would not have had our understanding of the infinite, but we must not underestimate how close to our understanding his generation came. He knew perfectly well, for example, the difference between a potential infinity (something in principle unlimited) and a completed infinity (such as the geometrical line). However, he did not conceive of the enumeration of an infinite set – so in Q7a4 he says that one must be able to enumerate a multitude and since infinity is not a number, you can’t have an infinite multitude. Now, although current physics does suggest that the universe is finite in extent, it is not beyond our comprehension to think of an infinite-in-all-directions steady-state universe with stars dotted about evenly. In such possible world, there would have to be infinitely many stars.

Does the failure of Aquinas’s argument make any difference to theology? I’m not sure that it does. He seems to be concerned in Q7 to “safeguard” the infinity of the infinite realm of God against any possible intrusion from the mundane created world and put yet more distance between God and creation. Does it matter if he hasn’t in this regard? Hasn’t he done enough already in showing the extent of God’s transcendence?

Perhaps one concern that Aquinas may have had was that if creation could be infinite-in-all-directions, how could God be “outside” creation? Where would there be for him to “fit”? Of course, one of the things that Aquinas was not aware of, that we are, is the idea of the “intrinsic geometry” of the universe (i.e. being a four-dimensional space time manifold); that the universe, even if infinite, can be conceived of as an object and that “placing” God as transcending space and time need not be problematical. Still, I’m not too sure of the likelihood of Aquinas having this concern. He certainly saw no problem with putting God “outside” of time; putting him outside of space too would surely come naturally. He was well beyond panentheism centuries before anyone had thought of it!

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