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A Survey of Predicativity Robin Adams

Summary: A Survey of Predicativity
Robin Adams
October 1, 2008
1 The Early Years
1.1 Russell and Poincar’e
The word 'predicative' first appeared in Russell's note On Some Di#culties in
the Theory of Transfinite Numbers and Order Types [Rus07]. Paradoxes such
as Russell's Paradox show that we cannot form the class {x | #(x)} for all
propositional functions #(x). Russell proposed we call #(x) predicative if it
defines a class and non­predicative otherwise, but did not o#er a criterion by
which we could decide which propositional functions are which.
A first such criterion was o#ered by Poincar’e in the third part of his paper Les
Math’ematiques et la Logique [Poi06]. He proposed the vicious circle principle:
``The definitions which ought to be regarded as non­predicative are those which
contain a vicious circle.'' (p. 1063) He indicated by example what he meant by
`vicious circle': in both the Richard paradox and the Burali­Forti paradox, we
define an aggregate E, and make use of E within its own definition.
Poincar’e proposed that definitions involving such a `vicious circle' are il­
legitimate: ``A definition containing a vicious circle defines nothing.'' [Poi06,
p. 1065]


Source: Adams, Robin - Department of Computer Science, Royal Holloway, University of London


Collections: Computer Technologies and Information Sciences