Consider the Axioms 2.1-2.4 of the Peano Axioms:

**Axiom 2.1.** 0 is a natural number.

**Axiom 2.2.*** *If n is a natural number, then n++ is also a natural number.

**Axiom 2.3.*** *0 is not the successor of any natural number; i.e., we have n++ ≠ 0 for every natural number n.

**Axiom 2.4.** Different natural numbers must have different successors; i.e., if n, m are natural numbers and n ≠ m, then n++ ≠ m++. Equivalently, if n++ = m++, then we must have n = m.

If we start from the number“0” to create a series of natural numbers, it is simple to find that any of these natural numbers can be represented completely by no more than three basic symbols, “0”, “( )” and “++”. However, assuming the above four axioms, there still exist at least two possibilities to create natural numbers.

(i) One way to create is by defining a symbol such as“0.5”, “1.1”, or “⊙” to be a natural number which distinguishes from any natural number denoted by “0”, “( )” and “++”. We can test the fact that the Axioms 2.1-2.4 still hold. First, the Axiom 2.1 certainly allow “0.5”, “1.1”, or “⊙” to be a natural number. Second, to make our new extended numbers compatible with the Axiom 2.2, we recursively define 0.5++ be the successor of 0.5, (0.5++)++ be the successor of 0.5++, and so forth. Also, we can replace 0.5 with 1.1 or ⊙ and proceed the same procedures. Regard the Axiom 2.3, we still set n++ ≠ 0 for every n. Though it is not necessary, we can also define n++ ≠ 0.5, n++ ≠ 1.1, or n++ ≠ ⊙ for every n. The Axiom 2.4 doesn’t make any contradiction when we bring in “0.5”, “1.1” or “⊙” and their corresponding successors to be natural numbers. We should note that we have defined 0.5 ≠ 0 in the very beginning. Then, we can define 0.5++ ≠ 0++, (0.5++)++ ≠ (0++)++, etc. Thus, we still satisfy the requirement of the Axiom 2.4 that different natural numbers must have different successors. Similarly, we can define 1.1++ ≠ 0++, (1.1++)++ ≠ (0++)++, …; we can also define ⊙++ ≠ 0++, (⊙++)++ ≠ (0++)++, … . Hence, the new extended numbers would follow the Axiom 2.4. Further, we can define 0.5, 1.1 and ⊙ to be three different natural numbers. By similar procedures shown above, we can define 1.1++ ≠ 0.5++, (1.1++)++ ≠ (0.5++)++, …, and ⊙++ ≠ 0.5++, (⊙++)++ ≠ (0.5++)++, …, and so forth.

(ii) One may define a different operation from increment to extend the natural number system. For instance, we can define a new operation such as “insert ⊙ forward”, then “⊙⊙”, “⊙⊙⊙”,… are also natural numbers. Similarly, we can define ⊙ ≠ ⊙⊙, ⊙⊙ ≠ ⊙⊙⊙, … .Through the same method shown in (i), we can create a new series of natural numbers from the start number such as “⊙⊙” , “⊙⊙⊙”.

However, the Axiom 2.5 of the Peano Axioms excludes the above two possibilities (i) and (ii) to create unusual natural numbers. It sums up that only those numbers denoted by “0”, “( )” and “++” can legally be natural numbers.

**Axiom 2.5.** Let P(n) be any property pertaining to a natural number n. Suppose that P(0) is true, and suppose that whenever P(n) is true, P(n++) is also true. Then P(n) is true for every natural number n.

It is the first post based on my own mathematical thinking. -Yandong Xiao

good job!