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Single Idea 21558
[filed under theme 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
]
Full Idea
Norms (containing one variable) which do not define classes I propose to call 'non-predicative'; those which do define classes I shall call 'predicative'.
Gist of Idea
'Predicative' norms are those which define a class
Source
Bertrand Russell (Difficulties of Transfinite Numbers and Types [1905], p.141)
Book Ref
Russell,Bertrand: 'Essays in Analysis', ed/tr. Lackey,Douglas [George Braziller 1973], p.141
Related Ideas
Idea 21559
We need rules for deciding which norms are predicative (unless none of them are) [Russell]
Idea 21568
A one-variable function is only 'predicative' if it is one order above its arguments [Russell]
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]
|
21558
|
'Predicative' norms are those which define a class
[Russell]
|
21559
|
We need rules for deciding which norms are predicative (unless none of them are)
[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]
|
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]
|
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]
|
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]
|