The concept of infinity immediately interests most students. That something can go on forever is quite astounding. Everyone who looks up to the night sky must ponder just how big the universe is. I suspect most people think it goes on forever, that it is infinite. Yet, the universe is finite. It is finite in time from its birth at the Big Bang until now, and it is finite in the distance it spans.
To study infinity, however, we don't look to the universe or to nature — we turn to set theory. In Faticoni's new book, The Mathematics of Infinity, we find a wonderfully lucid and detailed explanation of set theory to show us what infinity is and how to think about it and manipulate it.
This book begins with a simple introduction to sets such as elements of a set, and simple operations such as union and intersection. Each concept is clearly defined, explained, and illustrated. There is, to take just one, the definition of the cardinality of a set, say A, as "the sets X that can be matched element by element with A."
I have seen other books simply say the cardinality of a set is the number of elements in the set, which misses a subtlety. The text defines functions and mappings so we may count elements of a set and learn about countable and uncountable infinities.
By chapter three, we meet infinity. The author makes clear that big, no matter how large, is still not infinity. He illustrates countable infinity with Hilbert's hotel. This hotel has just one floor with an infinite number of rooms, labeled 1, 2, 3, etc. With this hotel as a guide, the author shows how subsets of an infinite set can still be infinite, and of course, have the same cardinality as the original set.
We learn that not all infinities are the same and find the proofs of the cardinality of the natural numbers, the cardinality of rational numbers, and the famous proof for the uncountability of reals. For a student new to these ideas, the presentation is excellent.
The ideas go deeper with the introduction of power sets, the set of all subsets of a set. Cantor's aleph nought makes its first appearance and "an infinite chain of infinite cardinals” comes to light. The arithmetic of cardinals is defined with many examples for illustration. The author takes care to explain all the steps to make them clear.
After cardinals, we meet ordinals. A cardinal denotes the size of set, but an ordinal is altogether different. It is a symbol not in a set but one that we can join, with the union operation, to the set to make this newly defined set larger than the first infinite set. Infinity now becomes even more interesting.
Here's how we discover ordinals. We stretch our thinking and view numbers as simply distinct symbols. They can indicate a counting, but they are now a way of writing different symbols or, if you will, elements of a set. Thus, 13 and 457 are symbols. There is also an ordering assigned to elements so that the successor to 13 is 14 and the predecessor to 457 is 456. These are not distances as a beginning student may think, they are simply an ordering. This ordering and concept of symbols of the set allows us to place a symbol beyond the end, so to speak, of the natural numbers. This new symbol is omega-nought. It succeeds every natural number and is the first infinite ordinal. From this first ordinal, we construct still larger sets, and come to the first uncountable ordinal.
We then learn of the Continuum Hypothesis and how mathematics can be independent of it. The text goes on to transfinite induction as well but the discussion is shallow.
The last two chapters are, frankly, out of place. They discuss prime numbers and some topics of logic and paradoxes. These parts of the book, unfortunately, lack the careful presentation that the earlier chapters have. For example, we are told of a 25-degree polynomial to generate prime numbers and the author invites the reader to program it on his desktop. But the parametric values we need for the polynomial to run a computer program are not given. The author touches on geometric series and the Riemann Zeta function but with little detail. The reader can say he has seen something of them, but nothing more.
This book gives a wonderful introduction of infinity and how infinity is a set theoretic idea built on symbols, subsets, and cardinality. I wish the book had stopped there. The last two chapters need not be part of it and the reader would not miss them. The book, though, is worthwhile for anyone beginning to think and learn about infinity.
David Mazel is an engineer in the Washington, DC, area. He received his bachelor’s degree from Old Dominion University and his Master’s and doctorate from Georgia Tech, all in electrical engineering. He is currently interested in billiards, cellular automata, and signal processing research.