Understanding of numbers-Natural Numbers

What is shu4 )? Or what is the natural number? For this question, most people will give an informal answer: 0, 1, 2,... is a natural number. This is what elementary school textbooks say. Note that I use "informal" to describe the above description of natural numbers. Strictly speaking, such a description cannot satisfy mathematicians. Because it is too general.

 

Mathematicians like "strict" things, which are also strict in constructing the beauty of the mathematical building. Strictly speaking, mathematics prefers to draw the most perfect conclusion from the least assumptions and the most rigorous proof. The above descriptions of natural numbers seem to introduce a large number of assumptions, which cannot satisfy mathematicians. Too many assumptions mean that the opportunities for the emergence of paradox are greater, and the possibility of building collapse is higher.

 

I understand two types of definitions of Natural Numbers: one is always trying to associate natural numbers with an external object. For example, the set base, the length of a straight line, and the object quality. The other is abstract, the so-called piano principle. The greatness of piano's principle lies in its attempt to reveal the essence of the number in a physical and chemical way. This means that when a number is transferred from one theory to another, it is no longer restricted by any external object, so that the number is unique in any entity and has its own meaning. Let's take a look at how the piano principle achieves this.

 

The piano principle assumes a basic number "0" (zero) and a basic operation "++" (ascending ). I cannot explain the rationality of the two basic components. Just as you cannot explain whether the fifth principle of Euclidean ry is true or not, the existence of these two components is a hypothetical premise. This ensures that the assumptions are minimized, because you do not need to perform operations such as 1, 2,..., addition, subtraction, multiplication, and division, so as to ensure the rationality of subsequent proofs as much as possible. This is important!

 

Based on 0 and ++, we now provide the First and Second Piano principles:

1. 0 is a natural number.

2. If n is a natural number, N ++ is also a natural number.

Justice 1 and 2 give us a series of natural numbers 0, 0 ++, (0 ++) ++, (0 ++) ++ ,... in addition, theorems 1 and 2 reveal an increasing relationship between natural numbers. It is worth noting that I have not introduced 1, 2, 3 ,... they appear a little later to avoid the reader's misunderstanding of piano. In this way, the meaning of N in principle 2 is enriched. It can be 0, 0 ++, (0 ++) ++ ,...

However, we cannot guarantee that the natural number is infinite. Because if (0 ++) ++ = 0, then the values 1 and 2 will fall into a terrible endless loop. Therefore, principle 3 emerged.

3. 0 is not the successor of any natural number, that is, for any natural number N, N ++ is not equal to 0.

This is not enough, because if any natural number and its successor (or successor ...) equal. For example, if 0 ++ = (0 ++) ++, the natural number will also fall into an endless loop, but this loop will not start from the beginning. We need principle 4.

4. different natural numbers have different successors, that is, if the natural number N is not equal to m, then n ++ is not equal to M ++, or if n ++ = m ++, then n = m.

Now the principle 1 ~ 4 fully describes an infinitely increasing set of Natural Numbers (the set is used here because the definition of the set can be independent of the natural number), but this is not enough.

 

In line with the principle 1 ~ 4. The numbers obtained are all natural numbers. This is true, but there is no reason to reject them that do not conform to the principle 1 ~ The number of 4 enters the natural number family, such as 0.5. Of course, this example is somewhat far-fetched, because we have not defined the real number 0.5 (the definition of the real number must be based on the proportion defined by the natural number), but at least it should be explained, we need another principle to ensure that all do not conform to the principle 1 ~ The number of 4 cannot be a natural number. The natural number only contains the number that increases from 0 through a certain number of times.

 

5. P (n) is an attribute of a natural number. If

(1) P (0) is true and

(2) If P (k) is true, P (K ++) is true.

If two conditions are true, P (n) is true for all natural numbers.

 

Principle 5 is the famous mathematical induction. The appearance of principle 5 ensures the necessity of principle 1 to 4, that is, it conforms to the principle 1 ~ 4 is a natural number.

Justice 1 ~ 5 perfectly defines an infinitely increasing set of natural numbers 0, 0 ++, (0 ++) ++, (0 ++) ++ ,..., it also reflects the abstract nature between natural numbers.

 

If 1 represents 0 ++, 2 represents (0 ++) ++ ,... we get the familiar natural number set 0, 1, 2 ,... however, we need to understand that except for 0, all symbols are just a representation of the abstract nature of natural numbers, and even include those that conform to 0. We need to separate the essence of a natural number from the symbols used to represent the essence. For example, we can also represent a natural number as 0, 1, 01, 10 ,... or 0, 1 ,..., a, B, C, D, E, F, 10 ,... or zero, one, two ,... and so on. When you really understand the separation of the essence and representation, you truly understand the beauty of piano's justice.

 

Santa

 

References: Chapter 2 of Tao zhexuan real analysis