This semester, we are taking a course on “Real Analysis.” The first chapter of Real Analysis covers sets. Regarding the cardinality of infinite sets, there are many conclusions that are counter-intuitive. The proof techniques for these conclusions are the core methods of set theory. However, I have found that although many conclusions contradict our intuition, upon careful reflection, they are not as “outrageous” as we might imagine. Yet, our current textbook, Real Analysis and Functional Analysis (by Cao Guangfu), does not use seemingly simple proofs. Instead, it employs relatively complex theorems, giving an impression of making things unnecessarily obscure.
I. The Set of All Real Numbers Cannot Be Put into One-to-One Correspondence with the Set of All Positive Integers
This is one of the fundamental conclusions in set theory. The proof is very simple: if the set of all real numbers could be put into one-to-one correspondence with the set of all positive integers, then the real numbers in (0,1) could also be put into one-to-one correspondence with the set of all positive integers. Represent all real numbers in (0,1) as infinite decimals without 0 as a recurring period (for example, 0.1 is represented as 0.0999\dots). Then, let a correspondence be: \begin{aligned} &a_1=0.a_{11} a_{12} a_{13} a_{14}\dots\\ &a_2=0.a_{21} a_{22} a_{23} a_{24}\dots\\ &a_3=0.a_{31} a_{32} a_{33} a_{34}\dots\\ &\dots\dots \end{aligned} where a_{ij} is any digit from \{0, 1, 2, \dots, 9\}, representing the j-th decimal place of a_i. Now, construct a number b=0.b_1 b_2 b_3 b_4\dots where b_i=1 if a_{ii} \neq 1, and b_i=0 if a_{ii} = 1. From the way b is constructed, it is clear that b is not among \{a_n\}, which contradicts the assumption.
The proof technique above is known as the “diagonal argument.” It is an extremely elementary and easy-to-understand technique. I remember being able to understand this proof back in middle school (even though I didn’t have a clear concept of sets at the time). I don’t know why our textbook uses the Nested Interval Theorem for such a high-level proof. Is it because once we have learned difficult things, we must use difficult proofs? Is this the only way to make the mathematics we learn seem “advanced”? I doubt I could have understood the Nested Interval Theorem in middle school.
II. The Union of Countably Many Continuum Sets is Still a Continuum
A continuum set is a set whose cardinality is equal to the cardinality of the set of real numbers. The cardinality of the set of real numbers is generally denoted as C. The proof of this proposition is very easy—it is even self-evidently true—yet for some reason, our textbook uses a large number of annoying symbols.
We know that the cardinality of the set of real numbers in [0,1) is C, the cardinality of the set of real numbers in [1,2) is C, the cardinality of the set of real numbers in [2,3) is C, and so on. Consequently, [0,1)\cup [1,2)\cup[2,3)\cup\dots=[0,\infty) is the set of all non-negative real numbers. Is its cardinality not C?
III. The Union of Countably Many Countable Sets is Still Countable
This is similar to the proof above. We know that the set of rational numbers in [0,1) is countable, the set of rational numbers in [1,2) is countable, the set of rational numbers in [2,3) is countable, and so on. Thus, [0,1)\cup [1,2)\cup[2,3)\cup\dots=[0,\infty) is the set of all non-negative rational numbers, which is naturally countable.
IV. \mathbb{R}^{\infty} is Still a Continuum
Here, \infty represents a countable infinity. This indicates that the set of real sequences \{x_1, x_2, x_3, \dots\} still has a one-to-one correspondence with the real numbers themselves, where x_i is any real number in (0,1). The proof here is not necessarily simple, but it is worth mentioning.
The proof is not very difficult, as long as one constructs an injection: \{x_1, x_2, x_3, \dots\} \mapsto (0,1) We only need to map the element \{x_1, x_2, x_3, \dots\} to y \in (0,1). The construction of y is as follows (let S_i(x) be the i-th decimal digit of x): S_{(2^i+2^{i+1} j)} (y)=S_{j+1} (x_{i+1}), \quad i,j=0,1,2,\dots This looks quite mysterious—how was it constructed? One only needs to write out the first few terms, and the reader will surely have an epiphany. This is also a very useful technique for proofs in set theory.
When reprinting, please include the original address of this article: https://kexue.fm/archives/2964
For more detailed reprinting matters, please refer to: Scientific Space FAQ