To show that the cardinality of the real numbers is equal to the cardinality of the set of all subsets of natural numbers (often denoted as 2^N or P(N)), denoted as |2^N| = |R| = C, we typically use Cantor's diagonal argument.
Cantor's diagonal argument proceeds as follows:
1. Assume that |R| = C.
2. Construct a list of real numbers, which we can think of as an infinite table.
3. Form a new number by taking the diagonal elements of the table and altering each digit (for example, by adding 1 and taking modulo 10).
4. This new number differs from each number in the list at least in one digit, forming a number that is not in the original list.
5. Thus, we have shown that there are more real numbers than those in the original list, contradicting the assumption that the original list contained all real numbers.
Therefore, |R| = C. This means that the cardinality of the real numbers is equal to the cardinality of the set of all subsets of natural numbers.
