Cantor's Theorem is a mathematical proof developed by Georg Cantor in 1891 that states that for any set, the number of elements in the power set (set of all subsets) is always greater than the number of elements in the original set.
Galileo's Paradox is a philosophical paradox raised by Galileo Galilei in the 17th century that questions the concept of infinity. It states that there are as many perfect squares as there are whole numbers, even though perfect squares are a subset of whole numbers and should therefore be fewer in number.
Cantor's Theorem is a direct response to Galileo's Paradox. It proves that even though there may be infinite sets, some of which are subsets of others, the power set of these sets will always be larger in size.
An example of a countable set is the set of all whole numbers (positive and negative). This set can be arranged in a one-to-one correspondence with the set of all even numbers, which is a subset of the whole numbers, showing that even though the even numbers are fewer in number, they still have the same cardinality as the whole numbers.
Cantor's Theorem is important in mathematics because it helps to understand the concept of infinity and the different levels or sizes of infinity. It also has applications in other areas of mathematics, such as set theory, topology, and analysis.