This theorem states that, 'for any set A, the power set of A,
which is all the subsets of A, has a greater cardinality than the
set of A.' Many examples can be used to show the trend, and then
through inductive reasoning, it can be concluded that the
cardinality of power sets will always be larger than their sets.
" S is n
and its power set P(S) is 2.
While this is clear for finite sets, no one had seriously considered
the case for infinite sets before the German mathematician Georg
Cantor—who is universally recognized as the founder of modern set
theory—began working in this area toward the end of the 19th
century.
^{n}"The 1891 proof of Cantor’s theorem for infinite sets rested on a version of his so-called diagonalization argument, which he had earlier used to prove that the cardinality of the rational numbers is the same as the cardinality of the integers by putting them into a one-to-one correspondence. The notion that, in the case of infinite sets, the size of a set could be the same as one of its proper subsets was not too surprising, as before Cantor almost everyone assumed that there was only one size for infinity. However, Cantor’s proof that some infinite sets are larger than others—for example, the real numbers are larger than the integers—was surprising, and it initially met with great resistance from some mathematicians, particularly the German Leopold Kronecker. Furthermore, Cantor’s proof that the power set of any set, including any infinite set, is always larger than the original set led him to create an ever increasing hierarchy of cardinal numbers, known as transfinite numbers. Cantor proposed that there is no transfinite number between the first transfinite number, or the cardinality of the integers, and the continuum ( c), or the
cardinality of the real numbers; in other words. This is now known
as the continuum hypothesis, and it has been shown to be an
undecidable proposition in standard set theory."^{4} |

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