**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:

1 = 01 = 001 = 0001 = ……01

2 = 02 = 002 = 0002 = ……02

3 = 03 = 002 = 0003 = ……03

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StepB2: Rewrite all natural numbers by opposite left-right direction. For example:

50792 could be rewrite as 29705

However, 50792 = ……0000050792

So it could be rewrite as 2970500000……

StepB3: After rewriting, all natural numbers can be listed:

N1 = 100000000000000……

N2 = 200000000000000……

N3 = 300000000000000……

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N50792 = 297050000000000……

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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:

real number rewritten real numbers

1st digit on the left of decimal point → 1st digit

1st digit on the right of decimal point → 2nd digit

2nd digit on the left of decimal point → 3rd digit

2nd digit on the right of decimal point → 4th digit

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For example:

…….00097531.24680000……… → 1234567890000000……….

…….00010267.57639000……… → 7567260319000000………..

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 definition of countable infinite 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:

x’s 1st digit on the left of decimal point → 1st digit

y’s 1st digit on the left of decimal point → 2nd digit

x’s 1st digit on the right of decimal point → 3rd digit

y’s 1st digit on the right of decimal point → 4th digit

x’s 2nd digit on the left of decimal point → 5th digit

y’s 2nd digit on the left of decimal point → 6th digit

x’s 2nd digit on the right of decimal point → 7th digit

y’s 2nd digit on the right of decimal point → 8th digit

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For example:

…051.370… + …062.480…I → 123456780000……….

Then we can finally get a bijection between complex number set and natural number set.