This was to be expected of the author, the British mathematician Marcus de Sautoy: his introduction to the properties of infinite sets discovered by Georg Cantor is devoid of technical jargon, generally understandable and highly entertaining.
What is the content provided in the short book? First, du Sautoy provides a brief historical review of the development of the count. Bone cracks (dating back tens of thousands of years) have been identified as counting marks, and the Lascaux Cave drawings also indicate early counting. The ancient Egyptians, Mayans and Babylonians not only developed the first signs of numbers, but also invented place value systems. The Mayans used the 20 as the basis, and the Babylonians the 60 – remnants of the number system can still be found today in the calendar.
Quantitative comparisons among natives
To explain the element scheme important to the rest of the book, the author finds an explanation somewhat elusive. The Angkamuthi (Aboriginal tribe in Australia) have numbers for 1, 2 and 3, and then only one for “many”. However, they can tell which pile of peaches and limes, both of which have the most fruit, have more. To do this, they remove one fruit from each of the pile one by one until one (at least) is gone.
Then du Sautoy comes to the gist of the book and describes the “Hotel Infinity” that David Hilbert created with Concierge Cantor at the reception. In an infinite number of hotel rooms with numbers 1, 2, 3, … can accommodate an infinite number of guests who traveled by bus and who have a natural number 1, 2, 3, … badge. This will also be possible if the guests only wear the even numbers 2, 4, 6, … on the pin. And the author can already conclude that the infinity of even numbers is only as large as the infinity of all natural numbers because they “can be perfectly assigned to each other”.
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