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 can be established, which is also called piano arithmetic system.

Piano's five axioms are described informally as follows:

① 1 is a natural number;

② Every certain natural number A has a certain successor number A'

，a '

It is also a natural number (the successor of a number is the number immediately after this number, for example, the successor of 1 is 2, the successor of 2 is 3, and so on. );

(3) If both B and C are successors of natural number A, then B = C；;

④ 1 is not the successor of any natural number;

⑤ Any proposition about natural number can prove that it is right for natural number 1 and right for natural number n'.

Then, this proposition holds true for all natural numbers. This axiom is also called inductive postulate, which ensures the correctness of mathematical induction.

Note: Inductive postulate can be used to prove that 1 is the only natural number that is not a successor, because the proposition is "n= 1 or n is the successor of other numbers", then the conditions of inductive postulate are satisfied.

If 0 is regarded as a natural number, then 1 in the axiom should be replaced by 0.

A more formal definition is as follows:

A Dai Deking-Piano Structure is a triple (X,

x，

f):

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, and 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, and the successor elements contain every 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 first inductive principle (mathematical induction), which is widely used.

_ _ _ _ _ _ _ _ _ _ No solution.