数系的构造 #
自然数、整数、有理数 #
通过皮亚诺公理假设自然数系的存在性
在自然数系中通过加减法运算得到整数系
欧几里得除法：
有理数的稠密性
有理数之间的间隙
实数 #
从有理数得到实数是从离散作为有理数序列的极限
十进制 #
在十进制中，所有的数字都由0,1,2,3,4,5,6,7,8,9来表示。
整数的十进制表示：
有理数的十进制表示：
实数的十进制表示：
一个实数可能有两个十进制表示
The construction of the real numbers aims to fill the infinite “gaps” or “holes” present in the rational number system (such as the absence of a rational square root of 2) by transitioning from a discrete system to a continuous one using limits. The logic of this construction proceeds through the following key concepts and steps:
1. Cauchy Sequences and Boundedness Because sequences of rationals might “want” to converge to a number that is not rational, the construction begins by identifying sequences that group closer together as they progress, without relying on the limit itself. This is formalized as a Cauchy sequence: a sequence of rational numbers $(a_n)_{n=0}^\infty$ is a Cauchy sequence if for every rational $\epsilon > 0$, the sequence is eventually $\epsilon$-steady, meaning there exists an integer $N \ge 0$ such that the distance $d(a_j, a_k) \le \epsilon$ for all $j, k \ge N$.
A fundamental property of every Cauchy sequence is that it is a bounded sequence. A sequence is bounded if there exists a rational number $M \ge 0$ such that $|a_i| \le M$ for all $i \ge 1$.
2. Formal Limits as Real Numbers Real numbers are defined directly from these sequences. A real number is constructed as an object of the form $\text{LIM}{n\to\infty} a_n$, which represents the *formal limit* of a Cauchy sequence of rational numbers $(a_n){n=1}^\infty$. To account for the fact that different sequences can target the same gap (e.g., $1.4, 1.41, 1.414…$ and $1.5, 1.42, 1.415…$), two real numbers $\text{LIM}{n\to\infty} a_n$ and $\text{LIM}{n\to\infty} b_n$ are defined to be equal if their underlying Cauchy sequences are equivalent, meaning they become eventually $\epsilon$-close to each other for every $\epsilon > 0$.
3. The Least Upper Bound Property Once the real numbers are defined and equipped with arithmetic operations and an ordering system, they can be shown to possess a defining characteristic that the rationals lack: the least upper bound property.
- An upper bound for a subset $E$ of real numbers is a real number $M$ such that $x \le M$ for every element $x \in E$.
- A least upper bound (or supremum) for $E$ is an upper bound $M$ such that any other upper bound $M’$ for $E$ must satisfy $M’ \ge M$.
The least upper bound property guarantees that every non-empty subset of real numbers that has an upper bound must have exactly one least upper bound. This property is what effectively “fills the gaps” in the real line, allowing us to rigorously prove the existence of numbers like $\sqrt{2}$.
4. Actual Convergence and Completeness With the real line fully constructed, one can replace formal limits with actual limits. Convergence of a sequence of real numbers $(a_n)_{n=m}^\infty$ to a real number limit $L$ means that for every real $\epsilon > 0$, the sequence is eventually $\epsilon$-close to $L$ (i.e., there exists an $N \ge m$ such that $|a_n - L| \le \epsilon$ for every $n \ge N$).
The logic of the construction concludes by proving that the formal limits used to build the reals are equal to their genuine limits, and by establishing the completeness of the real numbers: a sequence of real numbers is a Cauchy sequence if and only if it is convergent. This completeness confirms that the real number system has successfully eliminated the “holes” of the rational number system.