How to prove 1+ 1 = 2?

1+ 1=2 represents the history of axiomatization of natural numbers.

Axiomatization of natural numbers was first put forward by American mathematician Pierce in 188 1 year, and its definition is as follows:

Among them, x? Whether the last number is less than x.

Because subtraction and division are the inverse operations of addition and multiplication respectively (and are not close to natural numbers), only axiomatic addition and multiplication are needed.

According to the definition of Pierce's axiom, when 1+ 1 is x = 1, its value is the next number greater than y = 1, that is, 2.

Later, in 1888, the German mathematician Dai Dejin gave another set of axioms:

Let n be non-empty, and given an element e ∈ N in n, there is a mapping S: N → N on n, if:

Then triplet (n, e, s) is called natural number system, n is called natural number set, e is called initial element, and s is called subsequent element.

Dai Dejin, from a more essential level, axiomatized natural numbers. Through this axiom, we can define the addition and multiplication of natural numbers, which is equivalent to Pierce's axiom.

However, this axiom system is somewhat complicated (the mathematical logic language was just established at that time), so it has not attracted people's attention.

Followed by the next year, that is, 1889, the Italian mathematician piano, independent of Dai Dejin, published the Piano axiom:

Obviously, Piano axiom is a simplified version of Dai Dejin's axiom, so it is also called Dai Dejin-Piano axiom.

Using the Piano axiom, the addition of natural numbers is defined as follows:

Multiplication is as follows:

Use the above definition of addition to prove the problem of this question:

1 + 1 = 1 + 0? = ( 1 + 0)? = 1? = 2

The above axiomatic system is abstract, and there are different examples in different mathematical fields. Take piano's axiom as an example:

0 = 0

x？ = x + 1

0 = ?

x？ = x ∨{ x }

So there are:

1 = { 0 }, 2 = {0, 1}, 3 = {0, 1, 2}, ...

0 = λ .s λ。 z z

x？ = λ .x λ。 s λ。 z x s (s z)

So there are:

1 = λ .s λ。 z s z，2 = λ。 s λ。 z s (s z)，3 = λ。 s λ。 z s (s (s z))

Let c be a category and 1 be the terminating object of c, then define the category US? (c) The following:

What if we? An initial object (n, 0, s) can be found in (c), that is, for any object (x, 0? ，S？ ), there is a unique morphism u: (N, 0, S) → (X, 0? ，S？ ), it is said that C satisfies Piano's axiom. We? Every triple object in (c) is a piano axiom system.

It can be proved that these examples all meet the conditions defined by Piano's axiom, so these examples are well defined.

Because my math level is limited, mistakes are inevitable. Welcome the subject and the teacher to criticize and correct me! )

According to Goldbach's conjecture, I can't prove it

When 1 is only the definition of a number, 1+ 1=2 is no problem.

When 1 person is ten 1 person is not necessarily two absolute people!

For example, how many people are a good man and a murderer!

The answer is 1. Two people, two people and three people.

When a man lives with a woman. The answer is one person, two people, more people, no one!

A corrupt official and a mistress are caught, and in the end they may be a bunch of people or no one!

When you buy A Jin for fish and A Jin for chicken, the answer may be 2 jin, 2 jin 1 liang, 1 jin 82 liang, 1 jin 62 liang, or there is one or two left!

A lot of similar, netizen comments can be imagined!

This question involves piano's axiom.

The five piano axioms are:

(1)0 is a natural number;

(2) Every natural number A has a definite successor A', and A' is also a natural number;

(3)0 is not the successor of any natural number;

(4) Different natural numbers have different successors. If the successors of a and b are natural numbers c, then A = B；;

(5) If the set S is a subset of the set N of natural numbers, and two conditions are satisfied: χ, 0 belongs to S; 4. If n belongs to S, then the successor number of n also belongs to S; Then S is natural number set, and this axiom is also called inductive axiom.

The fifth rule of this axiom is disgusting. In view of your question, we can discuss the second one.

In the second axiom, it is assumed that the natural number 1 is followed by x', that is, 1+ 1 = x'. Then we define X' as 2, which means "1+1= 2"; Of course, it can also be defined as 0, but you need to find another name to replace the original 0, otherwise it will contradict axiom (3).

So 1+ 1 = 2 This is an artificial definition, which needs no proof and cannot be overturned. If 1+ 1 is not equal to 2, to put it bluntly, more than 99% theorems in the current mathematics field will all collapse and mathematics will start again.

But 1+ 1 has another meaning, which is the ultimate form of Goldbach's conjecture. No one can prove this conjecture at present, and the best proof at present is Chen Jingrun's 1+2, so Goldbach's conjecture 1+ 1 has not been solved, and I certainly can't provide any solution.

Since the use of Arabic numerals in the world, the first and most basic mathematicians have gradually emerged in the field of mathematics, such as addition, subtraction, multiplication, division and mixing. People constantly calculate these formulas. Make up a course. Receive basic mathematics education.

As for the era of advanced mathematics, all the great mathematicians in the world studied numbers. We don't understand.

Hehe, it's not what you think. The so-called "1+ 1" or "1+2" are just abbreviations. Goldbach conjectures that any even number greater than 6 can be expressed as the sum of two prime numbers, usually expressed as "1+ 1". China mathematician Chen Jingrun proved in 1966 that any sufficiently large even number is the sum of a prime number and a natural number, which can be expressed as the product of two prime numbers. Usually this result is expressed as "1+2". This is the best result of this problem at present. Please note that here, "1+ 1" is just a short name, not one plus one in the arithmetic sense. I'm afraid I won't write down Chen Jingrun's proof process here. Even if it is written here, how many people can understand it? If you mean "1+ 1" in the arithmetic sense, that is, how to prove that one plus one equals two, then I tell you, there is no need to prove it. One plus one equals two is the main postulate of mathematical axiom system. In other words, it is a postulate that one plus one equals two, which is self-evident and the premise of other mathematical theorems. So there is no question of how to prove that one plus one equals two. In addition, I want to remind you that what Chen Jingrun proved is not "1+ 1=2". This is common sense, don't joke. -If my answer helps you, please pay attention to me. Or have other questions, you can also pay attention to me and trust me privately.

Suppose 1+ 1 is not equal to 2.

Because 1+ 1 is not equal to 2.

And 2* 1 is binary addition (multiplication definition).

So 1+ 1 is not equal to 2* 1.

Because 1+ 1 is the addition of two ones (addition definition).

And 1*2 is also the addition of two ones (multiplication definition).

So 1+ 1= 1*2 because 1*2=2* 1 (multiplicative commutative law).

So 1+ 1=2* 1.

Contradict with the fourth line

So 1+ 1 equals 2.

1 plus 1 equals 2 without proof.

It is proved that the wrong view of "1 plus 1 equals 2" comes from a paper by Chinese mathematician Chen Jingrun. The topic of the published paper is "Representing a big even number as the sum of the products of a prime number and no more than two prime numbers", which is not what we think "1 plus 1 equals 2".

From 65438 to 0957, Chen Jingrun was transferred to the Institute of Chinese Academy of Sciences. As a new starting point, he studied harder. After 10 years of calculation,1May, 966, he published the paper "Representing even numbers as the sum of the products of a prime number and no more than two prime numbers".

The publication of the paper has been highly valued and praised by the world mathematics community and famous mathematicians. British mathematician Haberstein and German mathematician Li Xite wrote Chen Jingrun's paper into a math book called "Chen Theorem".

You think this 1+ 1 = 2 is fucking = 2? If you think it's not, then it's not. Why do you listen to (it) when some foreign bullshit scientists prove what theory and under what theory? Why do you ask others questions about 1+ 1 like a fool on the Internet? In reality, you live in a way of 1+ 1=2, instead of fucking going shopping. That is 1+ 1 = 2 yuan. You have to tell your theory to someone else's boss. Do you think you owe it? It makes you think. ? Or do you like reading more foreign books? Too lazy to fucking talk!

First, let me tell you the answer. This is the original definition of mathematics and does not need to be proved.

If you are still a student, congratulations on being a good student and ask why it is a good habit.