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Russel's Paradox (As Logicists both Frege and Russell agree that logic…
Russel's Paradox (
As Logicists both Frege and Russell agree that logic is foundational to Mathematics and that mathematics can be reduced to logic,
Hume's Principle(HP) is derived from Hume's original attempt to define number: “When two numbers are so combined as that the one has always an unit answering to every unit of the other, we pronounce them as equal”,
After realising the problem with HP, Frege sought an explicit account of number rather than an implicit one,
In 1902 Russell points to an inconsistency in Basic Law V: According to BLV, there is an extension(set) for any concept we can think of(Naive Comprehension Axiom). Therefore, consider the set of "all sets that are not members of themselves" #,
Russell believe's that rather than showing a problem with BLV, his paradox arises due to a fallacy. This fallacy is based in the use of impredicative definitions which lead to circularity. For him it is meaningless to talk about an extension that is or is not a member of itself.(Godel- a realist in ontology is not worried about impredicativity),
Frege finds Russell's Paradox devastating: “with the loss of my Rule V, not only the
foundations of my arithmetic, but also the sole possible foundations of arithmetic
seem to vanish”. #)