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AS ONE schoolboy succintly put it, 鈥淚nfinity is where things happen that don鈥檛.鈥� The infinite has posed huge challenges for mathematicians, who can鈥檛 live with it but can鈥檛 live without it, and philosophers, who can live without it but find it much more interesting to live with. Understanding the Infinite is a remarkable blend of mathematics, modern history, philosophy and logic, laced with refreshing doses of common sense. It is a potted history of, and a philosophical commentary on, the modern notion of infinity as formalised in axiomatic set theory. It also compares the orthodox mathematical view of infinity with some attractive alternatives, notably a theory developed by the Polish mathematician Jan Mycielski which replaces the infinite by the indefinitely large. If you think these are the same thing then you鈥檙e not a philosopher and you need this book.

Up to the 1870s, the classical approach to the infinite was that of potential infinity: not a thing but a process, one that can be continued as far as you wish. During the 1870s, however, the German mathematician Georg Cantor turned infinity into a thing by proposing a theory of actually infinite sets, in which an infinite process such as counting 1, 2, 3 鈥�, is not only considered to go on forever, but is dealt with as if it has finished and you can go beyond it. He adopted this position not out of a wish to make a nuisance of himself, but because various technical questions about the convergence of Fourier series, which mathematicians and physicists use to represent periodic functions, made it difficult to do anything else.

Using 鈭� as a symbol for whatever you get to after you have finished counting forever using the ordinary numbers 1, 2, 3, 鈥�, Cantor realised that you were then free to start again. He was led to a scheme of 鈥渢ransfinite ordinals鈥� that looks like this:

1,2,3, 鈥�, 鈭�, 鈭�+1, 鈭�+2, 鈭�++3, 鈥�, 鈭�.2, 鈥�, 鈭�.3, 鈥�, 鈭�², 鈥�, 鈭�³, 鈥�, 鈭�^鈭�, 鈥�

To convince yourself that this is a very natural idea, and not at all as outlandish as it seems, think of trying to count 鈥� in increasing order 鈥� the set of numbers:

0, 1/2, 3/4, 7/8, 15/16, 鈥�, 1

If 1, 2, 3, 鈥� going on forever is symbolised by 鈭�, then this set surely is 鈭�+1.

The British philosopher Bertrand Russell observed that the phrase 鈥渢he set of all sets that do not contain themselves鈥� is self-contradictory, implying that a set cannot necessarily be defined by stating a property that is satisfied by its members. Naive 鈥渋ntuitive鈥� set theory promptly bit the dust, and it has never been resuscitated. Cantor found a way to restrict the scope of the notion 鈥渟et鈥� to avoid Russell鈥檚 paradox. While Cantor鈥檚 transfinite ordinals increase indefinitely, no collection of them can decrease indefinitely: such collections are said to be 鈥渨ell-ordered鈥�. Cantor defined a set to be a collection of mathematical objects that can be well-ordered -鈥渃ounted鈥�, using his transfinite ordinals.

But Cantor ran into a technical difficulty. He could prove that the 鈥減ower set鈥� of any given well-ordered set 鈥� the collection of all of its subsets -corresponded to a bigger ordinal than the original set, provided it could be well-ordered. Unfortunately, he was unable to find any method for well-ordering the power set. In 1904 Ernst Zermelo, the German mathematician, obtained such a method by appealing to an apparently harmless assumption, the axiom of choice. This says that given any collection of sets, it is possible to choose exactly one member from each. This assumption is more subtle than it seems, and attempts to find 鈥済ood鈥� axioms for set theory 鈥� basic, 鈥渙bvious鈥� properties from which the entire theory can be derived 鈥� merely opened up a Pandora鈥檚 box of paradoxes and difficulties as each axiom led to a special case.

Understanding the Infinite is the best exposition of axiom schemes for set theory, and the reasoning behind those axioms, that I have ever encountered. It avoids philosophical hairsplitting by appealing to normal mathematical practice: 鈥渁ll else being equal, a theory that explains a practice [commonly used by professional mathematicians] will be superior to one that dismisses it鈥�. Because of this attitude, its philosophical probings help mathematicians 鈥� and interested onlookers 鈥� to understand what they have been doing, instead of trying to persuade them to do something else.

An unusual and fascinating feature is a description and analysis of Mycielski鈥檚 theory of the infinite. Some mathematical concepts can be easily specified, whereas others are much harder to pin down. Large numbers suffer from this difficulty. For example, while we can readily define the number 10鈭�10鈭�10鈭� (where 鈭� indicates exponentiation), there are many smaller numbers for which no compact notation exists. Indeed, most numbers smaller than this limit cannot be specified on a sheet of paper the size of the Universe, so their meaning becomes logically 鈥渇uzzy鈥�. Mycielski developed a coherent 鈥渇inite鈥� version of the foundations of mathematics by pursuing this idea. Understanding the Infinite is one of the few places where the nonspecialist can find an accessible description of what is involved. It is not an easy book, but it is an amazingly readable one given its difficult subject matter. Most of all, it is an eminently sensible book. Anyone who wants to explore the deep issues surrounding the concept of infinity, and is prepared to think hard or skip a few pages when the going gets tough, will get a great deal of pleasure from it.