Show that the power set of a set A, finite or infinite, has too many elements to be able t...

Show that the power set of a set A, finite or infinite, has too many elements to be able to be put in a one-to-one correspondence with A. Explain why this intuitively means that there are an infinite number of infinite cardinal numbers. [Hint: Imagine a one-to-one function 0 mapping A into (A) to be given. Show that ø cannot be onto (A) by considering, for each X ∊ A, whether X ∊ ø(x) and using this idea to define a subset S of A that is not in the range of ø.] Is the set of everything a logically acceptable concept? Why or why not?