Understanding the definition of natural numbers

Today, I carefully studied the definition of natural numbers that I have never really understood before in set theory and graph theory, and wrote down my own understanding.

What is a natural number? We all know that a natural number is a set {0, 1, 2, 3... N ,...}, this definition gives natural numbers a perceptual description that is sufficient in daily life. However, in mathematics, the definition of natural numbers was once the weakest point in the mathematical theory building. If the definition of natural numbers is not solved, many mathematical theories and methods related to natural numbers, such as mathematical induction, are doubtful. If this situation continues, mathematics is likely to become an empirical science, and even the entire mathematical building may crash due to a loophole in the definition of natural numbers.

Let's think about how natural numbers come from. 1, 2... n... these symbols correspond to a set of items in real life respectively. For example, the symbol 1 corresponds to an item, and the symbol 2 corresponds to two items ;... add an item to a set of N items to obtain n + 1 items. Remove an item from the set to obtain 0 items. That is to say, each natural number corresponds to the number of items in a pile in reality. Before there is no strict mathematical definition of natural numbers, the strictest way to explain to a person what is natural numbers should be to map a pile of actually existing item sets with symbols that represent natural numbers.

How can we get a strict mathematical definition of natural numbers? The so-called mathematical definition is to establish the definition of natural numbers on the basis of the set theory. A set is the most primitive concept in mathematics. It cannot be defined by other primitive concepts. Therefore, it has only properties and is not defined. However, since the concept of a set is simple enough, it is not difficult for people to understand its nature (even so, the set paradox once shocked the field of mathematics ). To define a natural number, we need to map it to a set. Such a set should have the nature of natural numbers:

1. It has an element;

2. All elementsYes and onlyOne successor (any element maps to another through a function), which also belongs to this set;

3,Yes and onlyAn element is not the successor of any element. It is.

So the design meets these three sets. Simply put, this set is as follows:

{Diameter, diameter +, diameter ++, diameter ++,...}, where a + = a margin {}.

In this case, we can call this set a natural number set.

Then, it is mandatory to map the regular symbols of natural numbers with the elements of this set, so that the natural numbers used in daily life get the meaning of the set theory, that is:

0 = Phi
1 = Phi + = 0 + = {0}
2 = Phi ++ = 1 + = {0, 1}
.
.
.
N = {, 2,..., n-1}