Chapter 52 Mathematics City (4)

Comrades, today is the foundation laying ceremony of Mathematics City. Many people asked me why I chose to hold this ceremony today. I can tell you the answer today, because 210 years ago, an Italian was born. The Italian was called Piano. He was a great man. Without him, many of the people here might not be qualified to sit here.

Why do you say this? Because the tree is thousands of feet tall and deep in the fertile soil. No matter how high our math level is, do you still remember our first math class? What is it taught? Isn’t it 11=2? 11=2 builds our most basic math knowledge. But I only realized that 11=2 is not that simple after a long time. Because it starts with the Piano axiom.

In 1889, based on the work of the mathematician Dedekin, Piano proposed an arithmetic axiom system in his book "Arithmetic Principles of Statement with a New Method". This axiom system has nine axioms, four of which are about "equal" and five are about numbers, and 1 instead of 0 is used as the basic concept. In later works, Piano modified this arithmetic system, removing the four axioms about "equal" and replacing 1 with 0 as the basic concept, constructing the Piano arithmetic axiom system that was used.

Human understanding of mathematics actually began. By the time of Piano, the Mathematics Building was actually very high. However, Piano found that the foundation of this Mathematics Building still needed to be strengthened. So he summarized the achievements of his ancestors, combined with his own opinions, and the opinions of his colleagues, and established the Piano arithmetic axiom system.

Although the mathematical language that describes this axiom system has undergone many changes, this system itself has been used to this day. Based on this natural number system based on axioms, the integer system can be obtained by introducing subtraction and division. Subsequently, the real number system is obtained by calculating the limits of the rational number sequence (proposed by mathematician Canto) or segmenting the rational number system (proposed by Dedekin). This axiomized real number system, together with Weirstrasse's contribution in the analyzing process of calculus (such as the ε-δ language in the limit definition), enables calculus, which has been used by humans for more than two hundred years, to be based on a solid foundation.

I, and most of the people here, are beneficiaries of this axiom system. Without this axiom system, we will find it difficult to learn calculus.

Note

These five axioms of Piano are described in a non-formal way as follows:

1    1 is a natural number;

2 Each determined natural number a has a certain successor number a', and a' is also a natural number (the successor number a' of the number a is the number (a1) immediately following this number, for example, 1'=2, 2'=3, etc.);

3If b and c are successors to the natural number a, then b   =     c;

4     1 is not a successor to any natural number;

5. If any proposition about natural numbers proves that it is right for natural number 1 and assumes that it is true for natural number n, it can prove that it is true for n', then the proposition is true for all natural numbers. (This axiom is also called inductive assumption, which ensures the correctness of mathematical induction)

Note: If 0 is also regarded as a natural number, then 1 in the axiom should be replaced with 0.

The more formal definition is as follows:

A Dedkin-Piano structure is a triple (x,    x,f) that meets the following conditions:

1    x is a set, x is an element in x, and f is a mapping from x to itself;

2    x    Not within the range of f;

3    f   For a single shot;

4 If a   is a subset of x and satisfies:     x belongs to    a, and if a         a, then       f(a) also belongs to a, then a   =x.

This axiom is combined with the basic assumptions about sets of natural numbers derived from Piarro's axiom:

1°                                                              

The set of 3° subsequent element map images is a true subset of p;

4° If any subset of p contains elements that are not successor elements and subsequent elements that contain each element in the subset, then this subset coincides with p.

These four assumptions can be used to prove many theorems that are common but do not know their origins!

For example: the fourth hypothesis is the theoretical basis for the first principle of induction (mathematical induction) that is widely used.

(Excerpt from "Baidu Encyclopedia")
Chapter completed!