Cantor digonalization method is used to prove that an open interval for real numbers $(0,1)$ is not countable. Since rational numbers also have decimal expansions, we could try to use this same method to prove that the set of rational numbers in $(0,1)$ is not countable. Explain why we could not conclude that the set of rational numbers in $(0,1)$ is uncountable using this method.
What i tried
Cantor diagonalizaton works by writing all the real numbers in the interval $(0,1)$ as an array of matrix and then finding a real number that does not belong to this set of array of matrix by making the diagonal entries of the real number different from that of the set of array of numbers in the matrix thus a contradiction. While for the set of rational numbers it can be expressed in the form of $\frac{a}{b}$ where gcd(a,b)=1, We do the same thing as for the rational numbers but i think this set of rational numbers somehow always tend to appear in the array of matrix no matter how we try thus this method of proof dosent apply for the case of rational number. But im unsure how. Could someone explain this to me. Thanks
Best Answer
Its not the process but the conclusion of the process that fails. When you write real numbers in the interval (0,1) as an array of matrix and then find a number by making the diagonal entries of the real number different from that of the set of array of numbers in the matrix, the resulting number is again a real number.
When you apply the same trick to the rational numbers the resulting number may not be rational. Remember being rational means the decimal expression either terminates or repeating. How do you ensure these properties in the resulting number?