In the last week's class, we learned more about countability and induction. We talked about the definition as countable for a set of number- basically iff it is listable (which means it can be described as a list for each item in its own position).
There is also a new concept called diagonalization. By using this concept, we can prove that the rational numbers are countable.e.g. The set of infinite number of floats in [0,1] can be put in a large list. Cantor was already proved that at least one number was missed when listing all numbers between 0 and 1, since the increment is keep decresing.
When using diagonalization proof, we add 1 to the tenth digit when digit is less than or equal to 5, minus 1 to the the tenth digit when digit is bigger than 5.And do the same thing to 100th digit and so on.After all, there will be a diagonal line in the matrix of digits.And a new number is created by this step.
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