Axiomatization of natural numbers was first put forward by American mathematician Pierce in 188 1 year, and its definition is as follows:
1 is the smallest number;
X+y, when x= 1, is the next number greater than y, otherwise, is the next number greater than X? The number of +y;
X×y, y when x= 1, and y+x in other cases? y；
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:
E is either the value of s or e? Ran;
S is injective, that is:? N，m∈N，(S(n)=S(m))？ (n = m)；
Inductive principle, that is, for any subset a? N, if e∈N and if n∈A, then a is n, that is:? Answer? N，( 1∈N)∧(( 1∈N)？ (S(n)∈A))？ (A=N)，
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.
Note: Here? Is it included? Is it really included? .
Followed by the next year, that is, 1889, the Italian mathematician piano, independent of Dai Dejin, published the Piano axiom:
0 is a natural number;
The successor number n of any natural number n? Or a natural number;
0 is not the successor of any natural number;
Two natural numbers are equal if and only if their successors are equal;
For subset A of natural number set, if 0∈N and n∈A, then n? ∈A is natural number set.
Obviously, Piano axiom is a simplified version of Dai Dejin's axiom, so it is also called Dai Dejin-Piano axiom.
Note: The earliest piano took 1 as the minimum natural number, and regarded the equivalence relation as a part of the axiom. The above is a later improved version.
Using the Piano axiom, the addition of natural numbers is defined as follows:
x+0=x
x+y？ =(x+y)？
Multiplication is as follows:
x0=0
xy？ =x+xy
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:
According to the oldest arithmetic:
0=0
x？ =x+ 1
According to set theory:
0=?
x？ = x ∨{ x }
So there are:
1={0},2={0, 1},3={0, 1,2}, ...
Hill odd number:
0=λ.sλ。 zigzag
x？ =λ.xλ.sλ。 Zxs (Shenzhen)
So there are:
1=λ.sλ。 zsz，2=λ.sλ。 zs(sz)，3=λ.sλ。 zs(s(sz))
According to category theory:
Let c be a category and 1 be the terminating object of c, then define the category US? (c) The following:
We? The object of (c) is the triple (x, 0? ，S？ ), where x is the object of c, 0? : 1→X and s? : X→X is a morphism of c;
We? (c) f:(X，0？ ，S？ )→(Y，0？ ，S？ ) is a C- morphism f:X→Y and satisfies: f0? =0? And fS? =S？ f，
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! )
Second, 1+ 1 = 2? Goldbach's Conjecture
1, many people don't understand why 1+ 1 = 2. Isn't this common sense?
However, there are many problems behind this question, which seem simple but wonderful. Let me answer why 1+ 1 = 2 needs to be proved and why it is so difficult to prove.
2. What is "1+ 1 = 2"
The so-called "1+ 1=2" actually refers to the Goldbach conjecture, which is called one of the three major mathematical problems in the modern world.
1742, Goldbach had a whim: "Any integer greater than 2 can be written as the sum of three prime numbers." But Goldbach himself could not prove it, so he wrote a letter to the famous Euler and put forward his conjecture, hoping that Euler would help him solve this problem.
However, faced with this wonderful conjecture, the great Euler could not give a reasonable proof until his death. Interestingly, hundreds of years have passed, but this problem that even primary school students can understand has stumped all mathematicians in the world.
3. Exciting facts
At present, the closest person to the perfect proof of 1+ 1 = 2 is Mr. Chen Jingrun, a famous mathematician in China. 1966, Chen Jingrun proved the theory of "1+2" in Goldbach's conjecture. This conclusion is called "Chen Theorem", which greatly advances the proof of Goldbach's conjecture.
Note: Before this, other mathematicians have gradually proved from "1+n" to "1+5", "1+4" and "1+3", also called screening method.
Chen Jingrun's "1+2" and "1+ 1" are only one step away. As long as the theory of "1+ 1" is proved, Goldbach's conjecture can come to a perfect end.
However, in fact, we are still far from the perfect proof of this problem.
4. Why is it difficult to prove?
Many people don't understand why Goldbach's conjecture is so great. In fact, the reason is that this conjecture can define almost all integers greater than 2. It is equivalent to telling the world, you see, all integers are made up of prime numbers.
And it's like when there is no microscope, someone suddenly puts forward that atoms are the smallest elements of all substances.
Proving Goldbach's conjecture is as difficult as proving that atoms make up everything without a microscope.
Step 5 write it at the end
Seeing many unfriendly answers to this question, I hope the subject will ignore it and pursue the truth is a great thing. But kindly remind the subject, don't try to prove 1+ 1 = 2 by yourself. Even if you claim that you have proved success, you will inevitably be called a folk subject.
6. This question concerns 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 both 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.
Conclusion: However, 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.
If you have any other views on the topic, please leave a message and discuss it together!