Ideas from 'Intros to Russell's 'Essays in Analysis'' by Douglas Lackey [1973], by Theme Structure

[found in 'Essays in Analysis' by Russell,Bertrand (ed/tr Lackey,Douglas) [George Braziller 1973,0-8076-0699-5]].

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5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / b. Cantor's paradox

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Sets always exceed terms, so all the sets must exceed all the sets

			Full Idea: Cantor proved that the number of sets in a collection of terms is larger than the number of terms. Hence Cantor's Paradox says the number of sets in the collection of all sets must be larger than the number of sets in the collection of all sets.
			From: Douglas Lackey (Intros to Russell's 'Essays in Analysis' [1973], p.127)
			A reaction: The sets must count as terms in the next iteration, but that is a normal application of the Power Set axiom.

5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / c. Burali-Forti's paradox

21553

It seems that the ordinal number of all the ordinals must be bigger than itself

			Full Idea: The ordinal series is well-ordered and thus has an ordinal number, and a series of ordinals to a given ordinal exceeds that ordinal by 1. So the series of all ordinals has an ordinal number that exceeds its own ordinal number by 1.
			From: Douglas Lackey (Intros to Russell's 'Essays in Analysis' [1973], p.127)
			A reaction: Formulated by Burali-Forti in 1897.