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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism

[maths entities only allowed if freshly defined]

19 ideas
Avoid non-predicative classifications and definitions [Poincaré]
To avoid vicious circularity Russell produced ramified type theory, but Ramsey simplified it [Russell/Whitehead, by Shapiro]
A one-variable function is only 'predicative' if it is one order above its arguments [Russell]
Russell is a conceptualist here, saying some abstracta only exist because definitions create them [Russell, by Bostock]
A set does not exist unless at least one of its specifications is predicative [Russell, by Bostock]
Vicious Circle says if it is expressed using the whole collection, it can't be in the collection [Russell, by Bostock]
We need rules for deciding which norms are predicative (unless none of them are) [Russell]
'Predicative' norms are those which define a class [Russell]
Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro]
Impredicative definitions are admitted into ordinary mathematics [Gödel]
If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock]
Predicativism makes theories of huge cardinals impossible [Bostock]
If we can only think of what we can describe, predicativism may be implied [Bostock]
The predicativity restriction makes a difference with the real numbers [Bostock]
The usual definitions of identity and of natural numbers are impredicative [Bostock]
If mathematics rests on science, predicativism may be the best approach [Bostock]
Some reject formal properties if they are not defined, or defined impredicatively [Shapiro]
'Impredicative' definitions refer to the thing being described [Shapiro]
Predicativism says mathematical definitions must not include the thing being defined [Horsten]