A new perspective of countable and uncountable innite set on Georg Cantor’s denition in set theory

Georg Cantor dened countable and uncountable sets for innite sets. Natural number set is dened as a countable set, and real number set is proven as an uncountable set by Cantor’s diagonal method. However, in this paper, natural number set will be proven as an uncountable set using Cantor’s diagonal method, and real number set will be proven as a countable set by Cantor’s denition. The process of argumentation provides us new perspectives to consider about the size of innite sets.

Introduction "In nite" is an unclear concept, and many scholars try to describe or de ne it. In set theory, the sets with in nite members are concerned and debated. Georg Cantor de ned countable and uncountable sets for in nite sets. The main concepts of Cantor's de nition for countable sets are: De nition 1: If the in nite members in a set can be listed by order, then the in nite set is a countable set.
De nition 2: According to the de nition given above, the natural number set is a countable set.
De nition 3: For any in nite set X, X is called countable if there is a bijection between X and the natural number set, or X is called uncountable.
The concepts are approved and applied by most scholars up to now. Under the concept and de nition, Georg Cantor declared that it is impossible to construct a bijection between natural number set and real number set. Furthermore, real number set is proven as an uncountable set by Cantor's diagonal method (1). The proof can be brie y described as follows: StepA1: Assuming that real number set is a countable in nite set.
StepA2: Under the assumption, the members in real number set can be listed by order. Any part of the members in real number set can be listed by order. Real numbers between 0 and 1 can be listed by order.
StepA3: Each real number can be represented by in nite decimal. For example: StepA4: Each real number between 0 and 1 can be represented by in nite decimal and can be listed. Mark them as s1, s2, s3, ………, sn, …… StepA5: When all real numbers between 0 and 1 are listed. We can construct a number S and let S differs from sn in its nth digit (Notice bold digits marked in StepA4): 1st digit of S cannot be 1 2nd digit of S cannot be 3 3rd digit of S cannot be 7 4th digit of S cannot be 8 StepA6: S is a real number. S is not any one real number listed above, since their nth digits differ.
StepA7: All real numbers between 0 and 1 are listed, so S should has been listed (Step A5). However, S is not any one real number in the list (Step A6). There is a contradiction under the assumption at Step A1. StepA8: The assumption "real number set is a countable in nite set" is wrong. Thus, real number set is an uncountable in nite set.
However, natural number set will be proven as an uncountable set using Cantor's diagonal method, and real number set will be proven as a countable set by Cantor's de nition in this paper. The results of argumentation will subversively change the concept of "in nite" in set theory, and the process of argumentation will provide us new perspectives to consider about the size of in nite sets.

Argument 1: Natural number set is uncountable
Natural number set could be proven as an uncountable set by the same demonstration program: StepB1: We know that nature number could be represented as different formats: StepB4: A rewritten natural number S can be constructed as: S differs from Nn in its nth digit StepB5: By the construction, S differs from each Nn, since their nth digits differ. According to the logic of Cantor's diagonal method, natural number set has been proven as an uncountable set.

Argument 2: Real number set is countable
Georg Cantor declared that it is impossible to construct a bijection between natural number set and real number set. However, I will construct a bijection between natural number set and real number set by following steps: StepC1 ~ StepC3: Rewrite all natural numbers as the same method described at StepB1 ~ StepB3 StepC4: Rewrite all real numbers as the sequence: Then we get a bijection between positive real number set and natural number set. Consider of positive number, negative number and zero, we get a bijection between real number set and integer set.
StepC5: According to the countable set theory, there is a bijection between integer and natural number set. So there is a bijection between real number set and natural number set. According to the de nition of countable in nite set, real number set is countable.
Moreover, it is easy to see that there is a bijection between complex number set and the natural number set by similar demonstration process. Each complex number could be written as x + yi, and both x and y are real numbers. We could rewrite complex number by following rules: Then we can nally get a bijection between complex number set and natural number set.

Discussion
Finding appropriate rewrite rules for any target numbers, and we can get one-to-one correspondence with the natural number set by similar demonstration process. I propose that we always can nd some rewrite rules to get a bijection between any two in nite set. In these demonstration processes, we nd that it is not necessary to distinguish countable or uncountable for any in nite set. It is nonsense to compare sizes of in nite sets because their members are all in nite. The size is de ned to describe nite numbers.
There is a new perspective of the contradiction in the Cantor's diagonal method. If we can list all real numbers, then we cannot construct the real numbers (StepA4 ~ A5) S and let S differs from sn in its nth digit. If we can construct the S, then we didn't list all real numbers. The contradiction at StepA7 makes Georg Cantor reject the assumption at StepA1, but it could make us reject the StepA4 ~ A5. The construction of S is after that all in nite numbers are listed. There should be a rational feature of in nite numbers: in nite numbers cannot be listed of all, or they are not in nite. In my opinion, the S constructed