Analysis (2 volume set) (Texts and Readings in Mathematics) by Terence Tao

By Terence Tao

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Thus, to define the natural numbers, we will use two fundamental concepts: the zero number 0, and the increment operation. In deference to modern computer languages, we will use n++ to denote the increment or successor of n, thus for instance 3++ = 4, (3++ )++ = 5, etc. This is a slightly different usage from that in computer languages such as C, where n++ actually redefines the value of n to be its successor; however in mathematics we try not to define a variable more than once in any given setting, as it can often lead to confusion; many of the statements which were true for the old value of the variable can now become false, and vice versa.

Thus to begin at the very beginning, we must look at the natural numbers. We will consider the following question: how does one actually define the natural numbers? (This is a very different question from how to use the natural numbers, which is something you of course know how to do very well. ) This question is more difficult to answer than it looks. , that a+b is always equal to b+a) without even aware that you are doing so; it is difficult to let go and try to inspect this number system as if it is the first time you have seen it.

0 Once we have a notion of addition, we can begin defining a notion of order. 11 (Ordering of the natural numbers). Let nand m be natural numbers. We say that n is greater than or equal to m, and write n ;:::: m or m ~ n, iff we have n = m + a for some natural number a. We say that n is strictly greater than m, and write n > m or m < n, iff n ;:::: m and n =f. m. Thus for instance 8 > 5, because 8 = 5 + 3 and 8 =f. 5. Also note that n++ > n for any n; thus there is no largest natural number n, because the next number n++ is always larger still.

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