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Single Idea 8746

[filed under theme 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism ]

Full Idea

Russell insisted on the vicious circle principle, and thus rejected impredicative definitions, which resulted in an unwieldy ramified type theory, with the ad hoc axiom of reducibility. Ramsey's simpler theory was impredicative and avoided the axiom.

Clarification

See Idea 8730 for impredicative definitions

Gist of Idea

To avoid vicious circularity Russell produced ramified type theory, but Ramsey simplified it

Source

report of B Russell/AN Whitehead (Principia Mathematica [1913]) by Stewart Shapiro - Thinking About Mathematics 5.2

Book Ref

Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.127

A Reaction

Nowadays the theory of types seems to have been given up, possibly because it has no real attraction if it lacks the strict character which Russell aspired to.

Related Idea

Idea 8730 'Impredicative' definitions refer to the thing being described [Shapiro]

The 19 ideas with the same theme [maths entities only allowed if freshly defined]:

18203 Avoid non-predicative classifications and definitions [Poincaré]
8746 To avoid vicious circularity Russell produced ramified type theory, but Ramsey simplified it [Russell/Whitehead, by Shapiro]
21568 A one-variable function is only 'predicative' if it is one order above its arguments [Russell]
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]
21559 We need rules for deciding which norms are predicative (unless none of them are) [Russell]
21558 'Predicative' norms are those which define a class [Russell]
8747 Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro]
10045 Impredicative definitions are admitted into ordinary mathematics [Gödel]
18131 If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock]
18134 Predicativism makes theories of huge cardinals impossible [Bostock]
18135 If mathematics rests on science, predicativism may be the best approach [Bostock]
18136 If we can only think of what we can describe, predicativism may be implied [Bostock]
18132 The predicativity restriction makes a difference with the real numbers [Bostock]
18133 The usual definitions of identity and of natural numbers are impredicative [Bostock]
13663 Some reject formal properties if they are not defined, or defined impredicatively [Shapiro]
8730 'Impredicative' definitions refer to the thing being described [Shapiro]
15370 Predicativism says mathematical definitions must not include the thing being defined [Horsten]