In logic, Richard's paradox is a semantical antinomy of set theory and natural language first described by the French mathematician Jules Richard in which year, Kurt Gödel specifically cites Richard's antinomy as a semantical analogue to his syntactical incompleteness result?
Richard's paradox In logic, Richard's paradox is a semantical antinomy of set theory and natural language first described by the French mathematician Jules Richard in 1905. The paradox is ordinarily used to motivate the importance of distinguishing carefully between mathematics and metamathematics. Kurt Gödel specifically cites Richard's antinomy as a semantical analogue to his syntactical incompleteness result in the introductory section of "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". The paradox was also a motivation of the development of predicative mathematics.
Galileo's paradox Galileo's paradox is a demonstration of one of the surprising properties of infinite sets. The ideas were not new with him, but his name has come to be associated with them. In his final scientific work, "Two New Sciences", Galileo Galilei made apparently contradictory statements about the positive integers. First, some numbers are squares, while others are not; therefore, all the numbers, including both squares and non-squares, must be more numerous than just the squares. And yet, for every square there is exactly one positive number that is its square root, and for every number there is exactly one square; hence, there cannot be more of one than of the other. This is an early use, though not the first, of the idea of one-to-one correspondence in the context of infinite sets.