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Full Idea
The Vicious Circle Principle says, roughly, that whatever involves, or presupposes, or is only definable in terms of, all of a collection cannot itself be one of the collection.
Gist of Idea
Vicious Circle says if it is expressed using the whole collection, it can't be in the collection
Source
report of Bertrand Russell (Mathematical logic and theory of types [1908], p.63,75) by David Bostock - Philosophy of Mathematics 8.1
Book Ref
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.225
A Reaction
This is Bostock's paraphrase of Russell, because Russell never quite puts it clearly. The response is the requirement to be 'predicative'. Bostock emphasises that it mainly concerns definitions. The Principle 'always leads to hierarchies'.
| 11064 | Classes can be reduced to propositional functions [Russell, by Hanna] |
| 6407 | The class of classes which lack self-membership leads to a contradiction [Russell, by Grayling] |
| 10418 | Type theory seems an extreme reaction, since self-exemplification is often innocuous [Swoyer on Russell] |
| 10047 | Russell's improvements blocked mathematics as well as paradoxes, and needed further axioms [Russell, by Musgrave] |
| 23478 | Type theory means that features shared by different levels cannot be expressed [Morris,M on Russell] |
| 21718 | Ramified types can be defended as a system of intensional logic, with a 'no class' view of sets [Russell, by Linsky,B] |
| 18126 | A set does not exist unless at least one of its specifications is predicative [Russell, by Bostock] |
| 18128 | Russell is a conceptualist here, saying some abstracta only exist because definitions create them [Russell, by Bostock] |
| 18124 | Vicious Circle says if it is expressed using the whole collection, it can't be in the collection [Russell, by Bostock] |