There are only seventeen Chinese characters in this definition.

So, why doesn't this Berry number exist?

Because, we assume that Berry number exists.

In other words, this Berry number needs at least eighteen Chinese characters to define.

So, this Berry number is defined by the above sentence.

That sentence has only 17 Chinese characters.

Therefore, this is self-contradictory-Berry number, which needs at least 18 Chinese characters to be defined, is defined by 17 Chinese characters.

So, let's just say there is no Berry.

So, without the Berry, is this the end?

Not exactly.

If the Berry number does not exist, that is to say, any integer can only be defined by no more than seventeen Chinese characters.

Because, if there are integers that need to be defined by more than 17 Chinese characters, then such integers naturally form a set-an integer set that cannot be defined by less than 18 Chinese characters.

However, as a non-empty subset of integer sets, this set must have a minimum value, and this minimum value is the smallest integer that cannot be defined with less than 18 Chinese characters, that is, Berry number.

However, we have just proved that Berry number does not exist, that is, there is no minimum value in this set that cannot be defined with less than eighteen Chinese characters. On an integer set, this means that the set is empty.

In other words, there is no "integer less than eighteen Chinese characters that cannot be defined".

However, we also know that there are infinitely many integers.

The number of Chinese characters is limited. Assuming H, how many integers can seventeen Chinese characters define? Less than H 17.

In other words, all the integer sets that can be defined by no more than 17 Chinese characters can only have H 17 elements at most, and there are infinite integers.

So, we have another contradiction-

We proved that Berry number does not exist, thus proving that there are no integers that need to be defined by eighteen or more Chinese characters, so all integers can only be defined by no more than seventeen Chinese characters, that is to say, the total number of integers can only not exceed H 17, but there are countless integers.

There is no problem in every link here, so who is wrong?

Let's look at another unrelated problem: Richard's paradox.

First of all, the above proposition itself should also be put into the proposition set given by 1, so in 2, give the proposition itself a serial number, and we assume it is m.

In other words, the above proposition is l _ m.

So, is l_m(m) true or false?

If l_m is a Richard proposition, then according to the definition of Richard proposition, we have l_m(m)=false, which means that the proposition l_m is not a Richard proposition.

Then, if l_m is not a Richard proposition, that is to say, l_m itself gives a wrong judgment on its serial number, that is, l_m(m)=false, then by definition, l_m is a Richard proposition.

Okay, this is a contradiction.

There is self-contradiction, which means there must be something wrong.

Based on this problem, modern mathematics puts forward the concept of "meta-mathematics"-mathematics discusses the properties of concrete mathematical objects, while meta-mathematics discusses the properties of mathematics, that is, "meta-mathematics is mathematical mathematics", and the subject of mathematics is studied by mathematical means.

So, why is this problem solvable in the meta-mathematical system?

Because "whether it is Richard's proposition" is the judgment of proposition, it belongs to the category of meta-mathematics, and we can call it "meta-proposition" or "quasi-proposition".

Here, propositions can only be judged by integers, and meta-propositions can only be judged by propositions.

The expression "l_n corresponds to n being Richard's proposition" is a meta-proposition, not a proposition. Since it is not a proposition, it can't appear in the proposition set mentioned by 1, so we can't assign a serial number to this meta-proposition, so there is no Richard paradox here.

Do you have the pleasure of being played?

Back to the Berry.

If the mathematical definition is to give a description of a number to uniquely determine this number, then let's see if the "definition" of "Berry number" conforms to this statement:

This statement is more like a proposition about definition: it doesn't give you what this number is, but tells you how to choose a definition from a bunch of definitions and take the number defined by this definition as the number represented by my definition.

Perhaps, we can think that this "definition" itself is not a mathematical category, but a meta-proposition of a meta-mathematical category or (like a brain hole) a "class definition".

Since we actually give a class definition, not a definition of Berry number, it is certainly not a "definition" of "you can't define less than 18 Chinese characters".

In other words, the class definition of Berry number cannot refer to itself, so it is impossible to construct the self-reference contradiction behind it.

So, is this the end of the problem?

Hehe, usually when I ask this question, the answer is obviously no.

Let's describe the previous problem in another way.

Suppose we now have a universal Turing machine U, which works on a selected ergodic enumerable language L.

Therefore, a string S can naturally be output by many Turing machines U without input parameters, that is, U () = S, and such U forms a set: U _ S.

Now, let's consider a special Turing machine in U_s: "shortest Turing machine" S', which satisfies Len(S').

Therefore, any string s can be "compressed" into S', and len(S') is called the "inherent length" of S.

Suppose Turing machine K is the Turing machine that finds the shortest Turing machine S' corresponding to any string S.

K is the length of k, that is, k=len(K).

Because L is a finite character set, the string S must be countable, and all the strings can be naturally arranged in an order according to the order of the characters in L and the length of the strings to form an ordered set, which contains all possible strings.

Next, we construct a Turing machine B, starting from the first character string in the above set, and continue to traverse until we find a character string that meets the following properties, and then stop and output the character string:

In other words, Turing machine B gives "the string with the smallest sequence number output by Turing machine with the length less than z".

Then, Turing machine B naturally has a length, marked B, and obviously has B >；; k .

Finally, we choose Z> B.

Well, we have reproduced the Berry number problem in another way.

If such Turing Machine B exists, the inherent length of the string S it outputs must not be less than z .. But B itself outputs S, and the length of B is less than z, so B should be the inherent length of S, which contradicts that the inherent length of S must not be less than z. ..

This is the uncountability of K- complexity proved by Charitin.

One way to solve this problem is that the length k of Turing machine K, which is likely to calculate the complexity of k, is infinite, so the above problem naturally does not exist, but this is not the focus of our discussion here.

The key point is, how did the Berry number problem I just tried to solve with the class definition of meta-mathematics reappear at the Turing machine level?

Moreover, if we slightly adjust the Richard paradox and reproduce it in the language of Turing machine, then we will be very pleased to find that this product is a downtime problem.

If the proposition is changed to Turing machine, and the integer is changed to data (Turing machine is also data before the great λ teaching), the output really means that it can be judged as downtime, and the output false means that it can be judged as non-downtime, then Richard paradox becomes a downtime problem.

However, just like the reappearance of Berry number problem at Turing machine level here, the meta-mathematical scheme that Richard paradox can solve is completely useless at Turing machine level.

I'm afraid its root lies in this point: at the Turing machine level, we can't tell what is the object in the mathematical field (definition and proposition) and what is the object in the meta-mathematical field (class definition and meta-proposition).

All the objects are Turing machines, so everything is mixed up.

The limitation that mathematical problems can't represent meta-mathematical objects has become air in the field of Turing machine, taboos have been broken, and the original solved problems have appeared again.

Is this a way to break away from convention and discover new fields, or go back to the original place and re-cage the original clear world with a layer of fog?

This is probably a matter of opinion. Bitch spilled hemorrhoids.

Today's bedtime story ends here, Oye ~ ~