Could anyone help me solve this propositional logic question

[deadcrrush]

Consider the infinitely large set of natural numbers {0, 1, 2, 3, ...}. Can there be another infinitely large set X that is bigger than the set of natural numbers, in the sense that it is not possible to pair each member of the set X with a unique natural number? Justify your answer.

I'm struggling to answer this question, any help appreciated.

 

[pka]

Consider the infinitely large set of natural numbers {0, 1, 2, 3, ...}. Can there be another infinitely large set X that is bigger than the set of natural numbers, in the sense that it is not possible to pair each member of the set X with a unique natural number?

You have asked about Cantor's Theorem: Every set is strictly dominated by its power set.
In other words, \(X \prec\mathscr{P}(X)\)
To prove that. Suppose that \(\Phi: X\to \mathscr{P}(X)\) is a bijection.
For each \([\forall T\in\mathscr{P}(X)][\exists t\in X]\{\Phi(t)=T\}\)
Now define a set \(A=\{x\in X: x\notin \Phi(x)\). Clearly because \(A\subset X\) then \(A\in\mathscr{P}(X)\).
Thus \((\exists a\in X)[\Phi(a)=A]\) So ask yourself, either \(a\in A~\vee~ a\notin A\) which is it?