Analysis

2.1 皮亚诺公理

Defition2.1.1

(informal)A natural number is any element of the set 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.

Axiom 2.1

0 is a natural number.

Axiom 2.2

If n is a natural number, then n++ is also a natural number

Axiom 2.3

0 is not the successor of any natural number; i.e., we have n++ 0 for every natural number n

Proposition 2.1.6

4 is not equal to 0.

Axiom 2.4.

Different natural numbers must have different successors; i.e., if n, m are natural numbers and , then . Equivalently, if , then we must have 

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也就是说这是利用逆否命题.

Proposition 2.1.8

6 is not equal to 2.

Example 2.1.9. (Informal)

Suppose that our number system N con￾sisted of the following collection of integers and half-integers:

Axiom 2.5 (Principle of mathematical induction数学归纳法).

Let be any property pertaining to a natural number n. Suppose that is true, and suppose that whenever is true, is also true. Then is true for every natural number n.

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这里的定理2.5只是一个公理模式，它可以构造(证明)很多公理，
1.$P_{0}$是真的;
2.当假设$P_{n}$是真的时，$P_{n++}$也是真的

we can use axiom2.5 to prove some proposition that just like proposition2.1.11

Proposition 2.1.11

Proposition 2.1.11. A certain property is true for every natural number n.

There are also some other variants of induction which we shall encounter later, such as backwards induction (Exercise 2.2.6), strong induction (Proposition 2.2.14), and transfinite induction (Lemma 8.5.15).

Axioms 2.1-2.5 are known as the Peano axioms for the natural numbers. (公理2.1-2.5被称为对于自然数的皮亚诺公理)They are all very plausible, and so we shall make assumption2.6

assumption2.6

(Informal) There exists a number system N, whose elements we will call natural numbers, for which Axioms 2.1-2.5 are true.

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A remarkable ac￾complishment of modern analysis is that just by starting from these five
very primitive axioms, and some additional axioms from set theory, we
can build all the other number systems, create functions, and do all the
algebra and calculus that we are used to.

注意: 有一个事实是每一个自然数都是有限的(用公理2.5证明)，但是整个自然数系(自然数的集合)确是无穷大的，不过存在包括无穷大的数系，比如基数系，系数系，p进数系(p-adics)，不过他们不遵循归纳法

Remark 2.1.15.

Historically, the realization that numbers could be treated axiomatically is very recent, not much more than a hundred years old. Before then, numbers were generally understood to be in￾extricably connected to some external concept, such as counting the cardinality of a set, measuring the length of a line segment, or the mass of a physical object, etc. This worked reasonably well, until one was forced to move from one number system to another; for instance, under￾standing numbers in terms of counting beads, for instance, is great for conceptualizing the numbers 3 and 5, but doesn’t work so well for ; thus each great advance in the theory of numbers - negative numbers, irrational numbers, complex numbers, even the number zero - led to a lot of unnecessary philosophical anguish. The great discovery of the late nineteenth century was that numbers can be understood abstractly via axioms, without necessarily needing a concrete model; of course a mathematician can use any of these models when it is convenient, to aid his or her intuition and understanding, but they can also be just as easily discarded when they begin to get in the way.

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简单地说，数系的存在本身就是人类理解的一个边界，他的每次扩充每次进步都是对人类理解和哲学的挑战，越来越难以理解，同时也是需要我们调整我们的理解和认知

Proposition 2.1.16 (Recursive definitions递归定义).

Suppose for each natural number n, we have some function from the natural numbers to the natural numbers. Let be a natural number. Then we can assign a unique natural number an to each natural number , such that and for each natural number .

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递归定义是一个关于自然数系很强大的工具，可以用数学归纳法证明，符合

Addition加法