How to prove 1+1 = 2?

Author:Shell net Time:2022.08.17

is this necessary?

If you look forward to a complete proof of Gothic Bach, I can only say that you are disappointed. The 1 and 2 I am talking about are pure natural numbers.

You start to disdain: 1 + 1 = 2 isn't it obvious? But have you considered, when we used to learn geometry, we always start with some axioms and gradually launch the required conclusions. However, this is not the study of algebra.

We have the addition table and multiplication table, and these tables have long been engraved in our minds. A system built by intuition does not seem to be credible. If the simple calculations such as 1 + 1 = 2 cannot prove, then all the results obtained through such operations are not credible, at least unscientific. It seems that we need to dig out something more basic than 1 + 1 = 2.

Picture source: Tuwa Creative

What is 1 and what is 2?

Before proof, we must first understand what natural number is and what is added. Similar to geometric axioms, we need to propose a few axioms, and then define the number of nature accordingly, and then define the addition.

Let's define the number of nature first. According to the significance of the number of nature (that is, the method of using the number of nature when the number of humans counts), it should start from one number, always count, and the count can be counted (that is, the natural number is infinite) Essence Based on this, we get the following axiom:

Governance 1. 0 is a natural number.

Truth 2. If n is a natural number, S (n) is also natural.

Here, S (N) represents N's "successor", that is, N to count another. That's right, what we usually say 0, 1, 2, 3, and ... is nothing more than a symbol of the above -mentioned mathematical object called "natural number". We use the symbol "0" to represent the original number of natural numbers, "1" to represent the subsequent S (0) of 0, and the subsequent S (1) of 1 is represented by the symbol "2", and so on.

However, only these two axioms are not completely described in a complete description of the number of nature, because it may not be the natural number system that meets these two. For example, consider the digital system composed of 0, 1, 2, and 3, where s (3) = 0 (that is, the number of the last number of 3 changes back 0). This does not meet our expectations for the natural number system, because it only contains limited number. Therefore, we must restrict the natural number structure:

Axiom 3. 0 is not the successor of any number.

However, the loopholes in it are not invincible. At this time, the following countermeasures cannot be ruled out: Digital System 0, 1, 2, 3, where s (3) = 3. It seems that the axioms we set up are not tight enough. We have to add another:

Truth 4. If n and m are naturally counted and n ≠ m, then s (n) ≠ s (m). In other words, the two natural numbers that are different, their respective successive successors are also two different numbers. In this way, the counter example mentioned above can be excluded, because 3 cannot be both the successor and the successor of 3.

Finally, in order to exclude some of the number of natural numbers (such as 0.5), and at the same time, in order to meet the needs of formulating the calculation rules for a while, we add the last axiom.

Truth 5. (Mathematics induction method) set P (n) a nature of natural number N. If P (0) is correct and assumes P (n) is correct, then P (s (n)) is also true. Then P (n) is correct for all natural numbers.

With the above efforts, we can define the natural number: there is a natural number system N, which is said to be natural, and only when these elements meet the axiom 1-5.

What is the addition?

We define that the addition is the operation of the following two rules:

1. For any natural count M, 0 + m = m;

2. For any natural number m and n, s (n) + m = s (n + m).

With these two additions that only depend on the "successor" relationship, the results of any two natural numbers can be determined.