Piano axiom, also called Piano postulate, is a system of five axioms about natural numbers put forward by mathematician piano. According to these five axioms, a first-order arithmetic system, also called piano arithmetic system, can be established.
Piano's five axioms are described informally as follows:
①0 is a natural number;
② Every certain natural number A has a certain successor A', and A' is also a natural number (the successor of a number is the number immediately after this number, for example, the successor of 0 is 1, the successor of 1 is 2, and so on. );
But these two axioms alone are not enough to describe natural numbers completely, because it may not be the natural number system that satisfies these two axioms. For example, consider a number system consisting of 0, 1, where the successor of 1 is 0. This is not in line with our expectation of the natural number system, because it only contains a limited number. Therefore, we should make some restrictions on the structure of natural numbers:
③0 is not the successor of any natural number;
However, the loopholes are hard to prevent. At this time, the following counterexamples are not excluded: digital system 0, 1, 2, 3, where the successor of 3 is 3. It seems that the axioms we set are not rigorous enough. We must add another one:
(4) If the successors of B and C are natural numbers A, then B = C；;
Finally, in order to exclude some numbers (such as 0.3) that should not exist in natural numbers, and to meet the needs of formulating operation rules later, we added the last axiom.
⑤ Any proposition about natural numbers holds true for all natural numbers. If it holds true for natural number 0, it can also prove true for natural number n'. (This axiom is also called inductive axiom, which ensures the correctness of mathematical induction. )
Note: Inductive postulate can be used to prove that 0 is the only natural number that is not a successor, because the proposition is "n=0 or n is the successor of other numbers", then the conditions of inductive postulate are satisfied.
If only positive integers are considered, 0 in the axiom will become 1.
A more formal definition
A Dai Deking-piano structure is a triad (x, x, f) satisfying the following conditions:
1, x is a set, x is an element in x, and f is the mapping of x to itself;
2.x is not in the range of f;
3.f is injective.
4. If A is a subset of X, X belongs to A, if A belongs to A, then f(a) also belongs to A, then A = X. 。
This structure is consistent with the basic assumption about natural number set derived from Piaro's axiom:
1 and p (natural number set) are not empty sets;
2. one of p to p-> is; One-to-one mapping of direct successor elements;
3. The set of subsequent element mapping images is the proper subset of P;
4. If any subset of P contains both non-successor elements and successor elements of each element in the subset, then this subset coincides with P. 。
Can be used to demonstrate many theorems that are common at ordinary times but don't know the source!
For example, the fourth hypothesis is the theoretical basis of the widely used first inductive principle (mathematical induction).
Definition of addition
We define addition as an operation that satisfies the following two rules:
1. For any natural number m, 0+m = m;
2. For arbitrary natural numbers m and n, n'+m = (n+m)'.
With these two definitions of addition that only depend on the "successor" relationship, we can determine the result of adding any two natural numbers.
Addition attribute 1+ 1=2.
1 + 1
= 0'+ 1 (according to the axiom of natural numbers)
= (0+ 1)' (according to addition definition 2)
= 1' (1 according to the definition of addition)
= 2 (according to the axiom of natural numbers)
associative law
In order to prove any one, the following proposition holds:
For any b and c, there is (a+b)+c=a+(b+c).
When a=0
(0+b)+c=b+c (addition definition 1)=0+(b+c) (addition definition 1), and the proposition holds.
Suppose the proposition is true for A, but true for A'
Let b and c have (a'+b)+c = (a+b)'+c = ((a+b)+c)' = (a+(b+c))' = a'+(b+c), and the proposition holds.
According to axiom 5, this proposition holds. Thus, the associative law a+(b+c)=(a+b)+c is obtained.
m'=m+ 1
When m = 0,0' =1= 0+1,the proposition holds. Suppose the proposition is true for m, and it is true for m', m' = (m+1)' = m'+1. According to axiom 5, this proposition is correct.
m+0=m
When m = 0, 1 is defined by addition. If it holds true for the natural number n, it can be proved that it holds true for n'. If it is axiomatic for the natural number 5, it is true.
Commutation law
Prove that the following propositions are true for any natural number n:
For any natural number m, m+n = n+m.
Now from the last paragraph, the proposition n=0 is true.
Suppose the proposition is right for proposition n and right for n'
m+n ' = m+(0+n)' = m+(0 '+n)= m+( 1+n)=(m+ 1)+n = m '+n =(m+n)' =(n+m)
Axiom 5, that is, the exchange law is established.
increase
Multiplication is an operation that satisfies the following two rules:
1. For any natural number m, 0 * m = 0.
2. For arbitrary natural numbers m and n, n' * m = (n * m)+m. 。
With these two definitions of addition that only depend on the "successor" relationship, we can determine the result of multiplication of any two natural numbers.
It can be proved that multiplication satisfies the following properties:
1. Multiplicative commutative law: a * b = b * a
2. Multiplication association law: a * (b * c) = (a * b) * c;
3. Multiplication distribution rate: a * (b+c) = a * b+a * c. 。
Subtraction and division
Integer is defined as natural number pair (a, b), and (a, b)=(c, d). If a+d=b+c, integer addition is defined as (a, b)+(c, d)=(a+c, b+d), and the inverse definition of (a, b) is (a). We have the concept of reciprocal. The sum of an integer and its reciprocal is 0, and the sum of 0 and any integer is itself. On an integer, a+(b) is defined as a+(the reciprocal of b). It can be verified that this definition is consistent with the commonly understood integer addition and subtraction.
The rational number is further defined as an integer pair [a, b], where b is non-zero. Definition [a, b]=[c, d] If ad=bc, define the multiplication of rational numbers as [a, b]*[c, d]=[a*c, b*d], and define the reciprocal of [a, b] as [define a-b as the antonym of a-b]. If [a, 1] is equal to a, it can be proved that the integer is a part of rational number, and the definitions of addition, subtraction, multiplication and division are consistent. So, on nonzero rational numbers, we have the concept of reciprocal. The product of nonzero rational number and its reciprocal is 1, 1.
If you are interested in this question, you can try to prove what is "provable" in the last article, or you can see how it is proved.
Real number, calculus
Piano's axiom was published by Italian mathematician piano in 1889. Although the mathematical language describing this axiomatic system has undergone many changes, the system itself has continued to this day. According to this axiomatic natural number system, the integer system can be obtained by introducing subtraction, and the rational number system can be obtained by introducing division. Subsequently, by calculating the limit of rational number sequence (proposed by mathematician Cantor) or dividing rational number system (proposed by Dai Dejin), a set of axiomatic real number system, and the contribution of Wilstrass in the process of calculus analysis at the same time (such as ε-δ language in the definition of limit), calculus that human beings have used for more than 200 years has been established on a solid foundation.
Algebraic structure
To sum up, our rational numbers and real numbers have four operations: addition, subtraction, multiplication and division. Are there other axiomatic systems and algebraic systems? The answer is yes.
Before answering this question, let's take a look at what an algebraic system is. First of all, what if there is only addition and subtraction? We can define Abel group as an algebraic system (G,+) with only addition and subtraction, where+satisfies:
1. associative law, (a+b)+c = a+(b+c);
2. Zero element, 0+a = a+0 = a;
3. Inverse number, each element A has an inverse number (-a), which satisfies the following conditions: A+(-a) = (-a)+A = 0;
4. commutative law, a+b = b+a.
On Abelian groups, subtraction can be defined as a-b=a+(-b).
Let's look at an example. We define G as a set of two elements {odd number, even number}. Let's define even+even = even, even+odd = even, odd+even = odd. We regard even numbers as 0, even numbers as even numbers and odd numbers as odd numbers. Then the addition and subtraction defined in this way also conforms to the basic operation rules of addition and subtraction. In other words, similarly, g can be defined as a set of n elements {multiple of n, multiple of n+ 1, ..., multiple of n +n- 1}. Such Abelian groups are mathematically called Zn groups. The Z2 group is the {odd, even} group mentioned above, which has parity and remainder, and 2 has no particularity compared with other numbers.
The definition of Abelian group can be obtained by removing the commutative law from the definition.
What if there are three operations of addition, subtraction and multiplication? A commutative ring is defined as (g,+,*), where (g,+) is an Abelian group, and (g, *) satisfies the associative law and the commutative law, with a distribution ratio: a * (b+c) = a * b+a * C, and if the multiplicative commutative law is removed, it is called a ring. For example, (finite decimal, addition, multiplication) forms a ring.
An algebraic structure with four operations of addition, subtraction, multiplication and division is called a field. Its formal definition is that commutative rings (g,+,*) are called domains. If there is a multiplication unit 1, then 1 * a = a * 1, and all elements A except 0 have reciprocal1/a.
For example, {odd, even} addition multiplication operation: even * even = even * odd = odd * even = even, odd * odd = odd, it becomes a commutative ring, and odd is the multiplication unit. This is called the binary number field. Generally speaking, Zn mentioned above can also form a commutative ring similarly, and form a domain when n is a prime number.
homomorphic
What if you use zero, one, two, three and so on to define another system? On the surface, 0 and 0, 1 and 1 seem to be completely different things. However, if we look at its essential connotation, zero and zero are just essentially the same things described in different languages. Mathematically speaking, it is reasonable to think that essentially the same thing is the same thing. In technical terms, it is "isomorphism".
Strictly speaking, two structures are defined as isomorphic if their elements correspond to each other and satisfy the same operation. For example, 1 corresponding to 1, 2 corresponding to 2, 1+ 1=2 corresponding to the past and then written as one plus one equals two, which is exactly the same as the original definition of addition.
The deeper concept is partial isomorphism, in other words, only when only one operation is considered, the two are consistent. For example, semi-integer = {0, 1/2, 1, 3/2, 2, 5/2, ...,-1/2,-65438+. Thus, in a popular geographical integer, 1. However, you may ask, 1/2*3/2=3/4? This actually shows that semi-integers can only be groups, not rings. It has only one addition structure, which is the same as the addition structure of integers. More generally, {0/n,1n, 2/n, ...-1/n, -2/n} also has the same addition as an integer.
In the last article, the semi-integer number 1 can be regarded as a literal value, which corresponds to the integer number 1, and its connotation corresponds to the integer number 2. This same but different nature, isomorphic but different structure, identity and difference contain profound philosophical thoughts. It has always been the core task of algebra to study whether algebraic structures are isomorphic and how many algebraic structures are different from each other.