What is Richard's Paradox?

好的，这个问题超有意思的！理查德悖论听起来很深奥，但它的核心思想其实可以用一个挺好玩的方式来理解。咱们一步步来拆解它。

What is Richard's Paradox?

Imagine we want to list all "numbers that can be described in Chinese."

Step One: Imagine an "All-Encompassing" Dictionary of Numbers

Let's compile a special dictionary. This dictionary won't contain Chinese characters in general, but rather entries for all numbers that can be clearly defined by a single sentence in Chinese.

For example, the dictionary would contain entries like these:

"one"
"pi"
"the square root of two"
"the smallest prime number divisible by two" (which is 2)
"the largest odd number less than ten" (which is 9)
...and so on, infinitely.

To make this dictionary more "scientific," we arrange all these Chinese descriptions in dictionary order (e.g., by Pinyin or stroke count) from top to bottom. This gives us an ordered list containing all numbers that can be defined in Chinese.

We number them:

The number corresponding to the first description (let's call it N1)
The number corresponding to the second description (let's call it N2)
The number corresponding to the third description (let's call it N3)
...
The number corresponding to the nth description (let's call it Nn)

This list is theoretically perfect; it contains every single number you can imagine and clearly describe in Chinese.

Step Two: The Birth of a "Troublemaker" Number

Now, based on that perfect list above, we're going to create a new, unique number. Let's call it "Richard's Number," R.

The creation rule is as follows:

We look at the 1st number N1 in the list, and take its 1st digit after the decimal point. Let's say it's 3. Then, for our new number R, its 1st digit after the decimal point will be a number different from 3, for example, 4.
Next, we look at the 2nd number N2 in the list, and take its 2nd digit after the decimal point. Let's say it's 1. Then, for our new number R, its 2nd digit after the decimal point will be a number different from 1, for example, 2.
Then, we look at the 3rd number N3 in the list, and take its 3rd digit after the decimal point. Let's say it's 4. Then, for our new number R, its 3rd digit after the decimal point will be a number different from 4, for example, 5.
And so on. For the nth number Nn in the list, we look at its nth digit after the decimal point, and then ensure that the nth digit after the decimal point of our new number R is different from it. (If the original digit is 1, we choose 2; if it's 2, we choose 1 – just make sure it's different).

This operation is called "diagonalization" in mathematics, but you don't need to worry about the name. You just need to know that through this "find the difference" method, we've created a brand new number R.

This new number R has a very important characteristic: it is absolutely not in our "all-encompassing" dictionary list!

Why?

It cannot be equal to the 1st number N1 in the list, because its 1st digit after the decimal point is different from N1's.
It cannot be equal to the 2nd number N2 in the list, because its 2nd digit after the decimal point is different from N2's.
...
It cannot be equal to the nth number Nn in the list, because its nth digit after the decimal point is different from Nn's.

Therefore, this "troublemaker" number R has indeed been excluded from our list.

Step Three: The Paradox Arrives!

Feeling a bit lost? Don't worry, the most crucial step is here.

You see, the problem lies here:

How did I just define that "troublemaker" number R? Didn't I describe it using a piece of Chinese text?

My description was: "A decimal number whose nth digit after the decimal point is different from the nth digit after the decimal point of the number corresponding to the nth description in our 'all-encompassing' dictionary list."

This sentence, though long, is indeed a clear Chinese description, isn't it?

So, here's the problem:

On one hand, according to our dictionary's compilation principle (to include all numbers that can be clearly defined in Chinese), R's Chinese description should be included in our dictionary. This means number R should be on that list.
On the other hand, based on our method of constructing R (the diagonalization 'find the difference' method), we have already proven that R is definitely not on that list.

This "troublemaker" number R should both be on the list and not be on the list.

Boom! This is Richard's Paradox. A logical contradiction arising from self-reference.

So, where exactly is the problem?

This paradox tells us that the problem isn't with mathematics itself, but with the natural language we use.

That seemingly perfect initial premise—"the list of all numbers that can be clearly defined in Chinese"—is itself problematic.

Why? Because the definition itself is too vague and contains "self-reference". When we define the "troublemaker" number R, we refer to the concept of "the list of all defined numbers" itself. This is like saying "This statement is false"; it refers to itself, leading to a logical loop.

To avoid such trouble, mathematics and logic later developed very strict "formal languages," which clearly distinguish between "language" and "metalanguage" (a language used to describe another language), thereby preventing the chaotic situation where things can easily refer to themselves.

Simple Summary

Richard's Paradox is a paradox about "definition."
It reveals that the idea of "a list containing all describable numbers" is inherently self-contradictory, by constructing a "describable number that is not on the list."
Its root lies in the ambiguity of natural language and the logical flaws introduced by "self-reference."

You can see it as a more complex version of the famous "Barber Paradox" (A barber in a village shaves only those who do not shave themselves. Question: Who shaves the barber?) in the field of mathematics. Their core is the logical conundrum triggered by "self-reference."