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I am trying to figure out what it means when they say something like "this definition generalizes over" such and such.

Example from https://www.iep.utm.edu/predicat/

The problem with the Russell class is said to be that its definition generalizes over a totality to which the defined class would belong.

Thank you!

user
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  • Usually that means that they are expanding a principle from the context it was first investigated in to other similar contexts. For example, one might make a claim about Saturn's rings from Cassini data, and then say "this claim generalizes over rings around any gas giant." – Ted Wrigley Apr 10 '20 at 16:14
  • @TedWrigley Thank you! So when used to explain why a definition is impredicative we would have: "The definition generalizes over a totality of things which includes the thing to be defined" means something like "I could apply the definition's criteria to every item in the totality, but the criteria also involve the totality"? – user Apr 10 '20 at 16:27
  • Yes, that seems right, though the term 'impredicative' creates some problems because it doesn't have a universally accepted definition. But, that detail you can figure out from context. – Ted Wrigley Apr 10 '20 at 16:31
  • @TedWrigley That was incredibly helpful. Thanks so much. – user Apr 10 '20 at 16:32
  • @MauroALLEGRANZA I have edited; thanks. – user Apr 10 '20 at 16:41
  • To say that "this definition generalizes over all natural numbers" means simply that the definition is about a natural number n that is defined with a statement involving an "universal quantifier" whose domain is (i.e. the quantifier "generalizes over") the set of all natural numbers. – Mauro ALLEGRANZA Apr 10 '20 at 16:59
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    See Universal generalization and Range of quantification. – Mauro ALLEGRANZA Apr 10 '20 at 17:13
  • @MauroALLEGRANZA Yes. That clears it up for me, I think. Is the following then a correct rephrasing of the definition of the Russel paradox set which explicitly indicates the kind of universal generalization/quantification you have mentioned? For every set X, X is a member of R iff X is not a member of itself. – user Apr 10 '20 at 18:48
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    Yes but... see Russell's Paradox: because the set R is defined as "the set whose members are exactly those objects that are not members of themselves" and thus its definition involves an universal quantifier ranging over all sets. – Mauro ALLEGRANZA Apr 10 '20 at 18:53

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