Question

Do all uncountable sets have the same cardinality?

Answer 1

the cardinality of the power set of real numbers is greater the cardinality of the real numbers

Answer 2

in general for any set A

|A | < | P(A)| < | P(P(A))| < ...

Answer 3

I thought real numbers were countable?

So this leads me to my next problem. My question is a multiple choice with more than one answer possible. I thought I had them all figured out but I don't.

So the question states: Which of the following are true?
1) All uncountable sets have the same cardinality. (<-- I think this is my problem)
2) Any set of real number is uncountable (FALSE b/c the set of integers are real and they are countable, right?)
3) The rational numbers form a countable set. (TRUE)
4) The power set of any infinite set is countable. (I thought this was FALSE b/c infinite sets can be either countable OR uncountable, correct?)
5) The real numbers form a countable set. (FALSE, correct?)