To prove that the power set of N has a higher cardinal number than N we must show that it is not possible to establish a one to one correspondence between.

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To prove that the power set of N has a higher cardinal number than N we must show that it is not possible to establish a one to one correspondence between the two: Imagine that all subsets of N are listed below:

To prove that the power set of N has a higher cardinal number than N we must show that it is not possible to establish a one to one correspondence between the two: Now number them:

The infinitely many natural numbers are listed here: x is under a member o is under a nonmember

Draw a diagonal Suppose we have a numbered list of sets of natural numbers

Reverse the elements on the diagonal

The green set is different from every set listed!

The green set is different from set 1 because the first element is different

The green set is different from set 2 because the second element is different

The green set is different from set 3 because the third element is different

The green set is different from set 4 because the fourth element is different

For every k, the k th element in the green set differs from the k th element in the set numbered k. It is impossible to count the power set of the set of natural numbers!

No matter how you try to line up the subsets on the right to count them, you can use this method to produce a set (like the green one) that is not on the list!