Hi, I met several times the expressions "predicative definition" and "unpredicative definiton" in texts about logic. What these expressions do mean ? I precise I'm a french student, thanks for your help.
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A definition of an object X is called impredicative if it quantifies over a collection Y to which X itself belongs (or at least could belong). The classic example is the set occurring in Russell's paradox, defined by "the members of X are all sets s that are not members of themselves". This quantifies over all sets, including X itself. But impredicative definitions occur (without paradox) in ordinary mathematics also. For example, one might define a real number r as the supremum of a set A that might have r itself among its members. Unraveling the definition of "supremum" we would find quantification over A (and indeed quantification over the set of all real numbers). Russell proposed to eliminate the set-theoretic and logical paradoxes by eliminating impredicative definitions, and "Principia Mathematica" (by Russell and Whitehead) develops an elaborate mechanism for this. Unfortunately, too much of ordinary mathematics was unprovable in that system, so Russell and Whitehead found it necessary to add the so-called axiom of reducibility, whose principal effect is to counteract the predicativity-enforcing mechanism and make impredicative mathematics available again. |
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I don't think "predicative" has a generally accepted precise definition, though roughly speaking it refers to definitions of objects that only depend on previously constructed objects. For example, the axiom of replacement in set theory is impredicative because it constructs new sets, but the construction involves quantifying over all sets, including the one you are trying to construct. The Feferman-Schutte ordinal is sometimes said to be the first impredicative ordinal (though I've never really understood why), in which case predicative mathematics consists roughly of theorems that only depend on transfinite induction up to smaller ordinals. |
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Wikipedia has a short article in French on this topic. |
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