history-page46 - Logical Issues in the 20th Century The...

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Unformatted text preview: Logical Issues in the 20th Century The Paradoxes of Naive Set Theory • Most boil down to sophisticated versions of the question: Is the sentence “This sentence is false.” true or false? • Recently, careful students of Cantor have made a strong case that Cantor knew of these and used careful phrasing to side-step them. ◦ Deiser (2004): “Long before presenting his definition of a ‘set’ [in 1895], Cantor was well aware that not every multitude could be considered a set.” • Russell’s Paradox ◦ If there is a Universal Set U , it is an element of itself. We can say it is self-contained . Consider the set S of all sets that are not self-contained. Is S self-contained? Responses to the Paradoxes • Rejection of an absolute Principle of Comprehension ◦ The Principle of Comprehension says that given any property there is a set consisting precisely of those things having that property....
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