关于皮亚诺公理（Peano axioms）定义自然数

因为想从头学习数学，最近在看陶哲轩的实分析，首先就是定义自然数：

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. Axiom 2.3. 0 is not the successor of any natural number; i.e., we have n++ = 0 for every natural number n.

Axiom 2.4. Different natural numbers must have different successors; i.e., if n, m are natural numbers and n = m, then n++ = m++. Equivalently2, if n++ = m++, then we must have n = m.

前四条看下来，有两个疑问：

"0"这个符号是自然数，但它是如何对应到我们所认知的 0 这个数量的
"++"这个动作为什么是+1，而不是+0.1，+0.77

作者显然知道我们会有这些疑惑，紧接着给出一个例子：

N := {0, 0.5, 1, 1.5, 2, 2.5, 3, 3.5,...}.

并且勾勒出目前只缺少一个公理：

What we want is some axiom which says that the only numbers in N are those which can be obtained from 0 and the increment operation - in order to exclude elements such as 0.5.

可是最后一个公理是数学归纳法原理：

Axiom 2.5 (Principle of mathematical induction). Let P(n) be any property pertaining to a natural number n. Suppose that P(0) is true, and suppose that whenever P(n) is true, P(n++) is also true. Then P(n) is true for every natural number n.

对 axiom 2.5 的补充说明是这样：

The informal intuition behind this axiom is the following. Suppose P(n) is such that P(0) is true, and such that whenever P(n) is true, then P(n++) is true. Then since P(0) is true, P(0++) = P(1) is true. Since P(1) is true, P(1++) = P(2) is true. Repeating this indefinitely, we see that P(0), P(1), P(2), P(3), etc. are all true - however this line of reasoning will never let us conclude that P(0.5), for instance, is true. Thus Axiom 2.5 should not hold for number systems which contain “ unnecessary ” elements such as 0.5. ——这里直将 0++ 赋值给变量“ 1 ”，1++赋值给变量“ 2 ”，并且直接认为变量"1"的值就是我们熟悉的数量 1，变量“ 2 ”的值就是我们熟悉的数量 2，由此证明自然数集合中不存在 0.5 这样的数。这样的推理科学吗？还有我上述的第一个疑问（"0"这个符号是自然数，但它是如何对应到我们所认知的 0 这个数量的）貌似也没有解决。