Comparing Russell´s Paradox, Cantor's Diagonal Argument And

Summary of Russell’s paradox, Cantor’s diagonal argument and Gödel’s incompleteness theorem
Cantor:
One of Cantor's most fruitful ideas was to use a bijection to compare the size of two infinite sets. The cardinality of is not of course an ordinary number, since is infinite. It's nevertheless a mathematical object that deserves a name, so Cantor represented it by the first letter in the Hebrew alphabet, , (pronounce "aleph") with a subscript of zero: ,
Cantors ingenious diagonal argument convinces us that there are so many real numbers that it is impossible for them all to be listed, even with a neverending list, and so they cannot be counted, even in unlimited time. Because the real numbers are associated with the points on a continuous line, their cardinality is called , the cardinality of the continuum.
His argument hence shows that represents a larger infinity than . Cantor then adapted the method to show that there are an infinite number of different infinities, each one surprisingly bigger than the one before.
Today this amazing conclusion is honoured with the title Cantor's theorem, but during his times most mathematicians did not understand it.
He used a generalized version of his diagonal argument to then prove that; for every set Q the power set of Q, i.e., the set of all subsets of Q (here written as P(Q)), is larger than S

And can link it to Dr. Kurt Godel’s work. When this came up, Russell after finding the hole in the previous set theory, claimed to fix it using his method of ‘types’. But then Gödel, in one of the most ingenious moves in the history of math, converted the paradox into a mathematical formula and proved that any statement requires an external observer and ended up proving russell’s theory wrong. His landmark discovery, as powerful as anything Einstein developed, was a devastating blow to the positivism of his