By Terence Tao
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Additional info for Analysis I (Volume 1)
This closes the induction, and thus P(n). is true for all numbers n. D Of course we will not necessarily use the exact template, wording, or order in the above type of proof, but the proofs using induction will generally be something like the above form. 15). 5 are known as the Peano axioms for the natural numbers. 6. 5 are true. We will make this assumption a bit more precise once we have laid down our notation for sets and functions in the next chapter. 12. We will refer to this number system N as the natural number system.
0 is a natural number. 2. If n is a natural number, then n++ is also a natural number. 2, we see that (0++ )++ is a natural number. 1. 3. We define 1 to be the number 0++, 2 to be the number (0++ )++, 3 to be the number ((0++ )++)++,etc. (In other words, 1 := 0++, 2 := 1++, 3 := 2++, etc. 4. 3 is a natural number. Proof. 1, 0 is a natural number. 2, O++ = 1 is a natural number. 2 again, 1++ = 2 is a natural number. 2 again, 2++ = 3 is a natural number. D It may· seem that this is enough to describe the natural numbers.
More generally, it is a fact (which we 28 2. The natural numbers shall prove shortly) that a+ b = b + a for all natural numbers a, b, although this is not immediately clear from the definition. ). Right now we only have two facts about addition: that O+m = m, and that (n++) +m = (n+m)++. Remarkably, this turns out to be enough to deduce everything else we know about addition. We begin with some basic lemmas4 . 2. For any natural number n, n + 0 = n. Note that we cannot deduce this immediately from O+m because we do not know yet that a + b = b + a.
Analysis I (Volume 1) by Terence Tao