By Terence Tao

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Additional info for Analysis (2 volume set) (Texts and Readings in Mathematics)

Example text

Then it gives a single value to 0-n++, namely 0-n++ := fn(an). ) This completes the induction, and so an is defined for each natural number n, with a single value assigned to each an. D 3 Strictly speaking, this proposition requires one to define the notion of a function, which we shall do in the next chapter. However, this will not be circular, as the concept of a function does npt require the Peano axioms. 12. 2. Addition 27 Note how all of the axioms had to be used here. In a system which had some sort of wrap-around, recursive definitions would not work because some elements of the sequence would constantly be redefined.

36 Proof. 5. 2. 11 (Exponentiation for natural numbers). Let m be a natural number. To raise m to the power 0, we define m 0 := 1. Now suppose recursively that mn has been defined for some natural number n, then we define mn++ := mn x m. 12. Thus for instance x 1 = x 0 x x = 1 x x = x; x 2 = x 1 x x = x x x; x 3 = x 2 x x = x x x x x; and ·so forth. By induction we see that this recursive definition defines xn for all natural numbers n. 10. 1. 2. 2. 3. 3. 5. 4. Prove the identity (a+b) 2 numbers a, b.

The Pean? axioms 17 However, we shall stick with the Peano axiomatic approach for now. How are we to define what the natural numbers are? 1. (Informal) A natural number is any element of the set N := {0,1,2,3,4, ... }, which is the set of all the numbers created by starting with 0 and then counting forward indefinitely. We call N the set of natural numbers. 2. In some texts the natural numbers start at 1 instead of 0,. but this is a matter of notational convention more than anything else. In this text we shall refer to the set { 1, 2, 3, ...