The main body of this chapter consists of a parade of short sections each of which concerns itself with some particular infinite set, infinite process, or point of view or technique for controlling infinity. We have tried to treat each of these separately, as much as possible; however, the first section begins with three infinities!
Natural Numbers
At the head of the parade of infinities, set off from the others, come the ordinary numbers. We are not going to be able to construct infinite sets, or prove anything significant about them, unless somehow we start with at least one infinite set, already constructed for us. Such a set exists, namely the ordinary numbers: 1, 2, 3, 4, 5, 6, … and so on. Concerning these, often called natural numbers, we are free to assume the following facts:
a) Each natural number has an immediate successor, so that the procession continues without end.
b) There is no repetition; each number is different from all the preceding numbers.
c) Every whole number can be reached in a finite number of steps by starting at 1 and counting up, one at a time, through the line of successors.
Discussion of Sequences
Since all numbers in the set of natural numbers cannot be written in a finite time we use “…”, that is, suspense dots or iteration dots, usually just three of them; they correspond to the words “and so on” or “et cetera”, or “and so forth”.
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