Is this necessary?
If you were expecting a complete proof of Goldbach’s conjecture here, all I can say is you’re disappointed buddy. The 1 and 2 I’m talking about are pure natural numbers.
You’re starting to dismiss it: Isn’t it obvious that 1 + 1 = 2? But have you ever considered that when we were studying geometry in the past, we always started with some axioms and gradually deduced the necessary conclusions. However, this is not the case with the study of algebra.
What we have are addition and multiplication tables, and these tables have long since become the intuition of calculation. A system built on intuition doesn’t seem very plausible. If even a simple calculation like 1 + 1 = 2 cannot be proved, then all the results obtained through such calculations are unreliable, at least unscientific. It seems that we need to dig into something more fundamental than 1 + 1 = 2.
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What is 1 and what is 2?
Before the proof,first we need to understand what is a natural number and what is addition. Similar to the axiomatic theoretical system of geometry, we need to propose several axioms, and then define natural numbers based on them, and then define addition.
First define the natural numbers. According to the meaning of natural numbers (that is, the way humans use natural numbers when counting), it should start with a number and count up, and you can count as many as you want (that is, natural numbers). There are unlimited). From this we get the following axioms:
Axiom 1.0 is a natural number.
Axiom 2. If n is a natural number, then S(n) is also a natural number.
Here, S(n) represents the “successor” of n, that is, one more number up from n. That’s right, what we usually call 0, 1, 2, 3, … is nothing more than a symbol representing the above-mentioned mathematical objects called “natural numbers”. We use the symbol “0” for the original natural number, “1” for the successor S(0) of 0, and the symbol “2” for the successor S(1) of 1, and so on.
However, these two axioms alone are not enough to describe the natural numbers, because it may not be a natural number system that satisfies these two axioms. For example, consider the number system consisting of 0, 1, 2, 3, where S(3) = 0 (i.e. the number after 3 becomes 0). This is not what we would expect from a natural number system, since it contains only finite numbers. Therefore, we have to make another restriction on the structure of natural numbers:
Axiom 3. 0 is not the successor of any number.
But the loopholes in it are hard to prevent. At this time, the following counterexample cannot be ruled out: the number system 0, 1, 2, 3, where S(3) = 3. It seems that the axioms we set are not rigorous enough. We have to add one more:
Axiom 4. If n and m are both natural numbers and n ≠ m, then S(n) ≠ S(m). That is to say, for two natural numbers that are different from each other, their respective successors are also two different numbers. In this way, the counterexample mentioned above can be ruled out, because 3 cannot be both the successor of 2 and the successor of 3.
Finally, in order to exclude some numbers that should not exist in natural numbers (such as 0.5), and also to meet the needs of formulating arithmetic rules later, we add the last axiom.
Axiom 5. (Mathematical induction) Let P(n) be a property about the natural number n. If P(0) is correct, and if P(n) is assumed to be correct, then P(S(n)) is also true. Then P(n) is true for all natural numbers n.
With the above efforts, we can define the natural number system as follows: There is a natural number system N, and its elements are called natural numbers if and only if these elements satisfy axiom 1 – 5.
What is addition?
We define that addition is an operation that satisfies the following two rules:
1. For any natural number m, 0 + m = m;
2. For any natural numbers m and n, S(n) + m = S(n + m).
With these two definitions of addition, which only depend on the “successor” relationship, the result of adding any two natural numbers can be determined.
How to prove 1+1=2?
At this point, we can prove that 1 + 1 = 2:
1 + 1
= S(0) + 1 (according to the axioms of natural numbers)
= S(0 + 1) (2 is defined by addition)
= S(1) (according to plusLaw Definition 1)
= 2 (according to the axioms of natural numbers)
In fact, according to the definition of addition, we can not only prove every addition equation, but also further prove general laws such as the associative law of addition and the exchange rate of natural numbers. Similar to the definition of addition, the multiplication of natural numbers can also be defined and the associativity, exchange rate and distribution rate of multiplication can be proved accordingly. If you are interested in this issue, you can take a look at the reference .
Seeing this, I wonder if you will feel a sense of relief. It turns out that everything we know about mathematics, and everything we know about the world, is not based on intuition, but is deduced by rational methods under the condition of accepting several axioms . At the same time, you may still feel a sense of freedom: just as you can construct your own geometric system without accepting Euclid’s axioms, you can also build your own system of numbers without accepting the above axioms. . You can build an infinite number of weird systems. But if it is to explain nature, at least from the current point of view, the existing set is better.
Some historical background
The Axioms 1 – 5 mentioned above are the famous Peano axioms, which were published in 1889 by the Italian mathematician Peano. Although the mathematical language used to describe this system of axioms has undergone many changes, the system itself has continued to this day.
According to this natural number system based on axioms, the integer system can be obtained by introducing subtraction, and then the rational number system can be obtained by introducing division. Subsequently, the real number system is obtained by calculating the limit of the rational number sequence (proposed by the mathematician Cantor) or dividing the rational number system (proposed by Dedekind). This system of axiomatic real numbers, together with the contribution of Weierstrass in the process of analyzing calculus at the same time (such as the ε-δ language in the definition of limits), has made calculus that has been used by humans for more than two hundred years. Learning can be built on a solid foundation .
 Analysis [M]. Terence Tao
 Introduction to the History of Mathematics (Second Edition) [M]. Li Wenlin
 A History of Mathematics, an Introduction (Second Edition) [M]. Victor J. Katz
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