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The concept of cardinality in set theory is akin to the idea of size when referring to sets. It indicates the number of elements in a set. For finite sets, cardinality is simply the count of elements. However, for infinite sets, things get a bit trickier. Two sets are considered to have the same cardinality if there exists a bijective function between them—a one-to-one mapping.In the context of Cantor's Theorem, we are dealing with the cardinality of a set, \(S\), and its power set, \(\mathscr{P}(S)\). Intuitively, the power set \(\mathscr{P}(S)\) (which includes all possible subsets of \(S\), even the empty set and \(S\) itself) tends to have a greater "size" or cardinality than \(S\) itself, which leads to the heart of the problem: proving these cardinalities are not the same.

Proof by contradiction is a logical method often used in mathematics to prove a statement true by proving that assuming the opposite leads to a contradiction.In Cantor's Theorem, the goal is to show that the set \(S\) and its power set \(\mathscr{P}(S)\) do not share the same cardinality. To do this, we assume the opposite: that a bijective function \(f: S \to \mathscr{P}(S)\) exists, mapping each element of \(S\) to a unique subset.However, the contradiction arises when defining a new set \(A\) consisting of elements that are not contained in their own mapping, i.e., \(A = \{x \in S | x otin f(x) \}\). This leads to unsolvable contradictions when checking whether an element can or cannot be part of \(A\), proving our initial assumption of bijection to be false.

The power set of a set \(S\), denoted as \(\mathscr{P}(S)\), is the set of all possible subsets of \(S\). If \(S\) has \(n\) elements, its power set will have \(2^n\) elements. This is because each element can either be included or excluded from a subset.The power set is essential in Cantor's Theorem because it encapsulates the concept that even for infinite sets, the power set will always have a strictly bigger cardinality than the original set. In other words, \(\mathscr{P}(S)\) is always larger than \(S\), as proven by the contradictions revealed when assuming a bijection is possible.These ideas together lead to one of set theory's most intriguing insights: no set can map in a one-to-one manner with its power set, emphasizing the hierarchy within infinity itself.