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Single Idea 18133
[filed under theme 6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
]
Full Idea
The predicative approach cannot accept either the usual definition of identity or the usual definition of the natural numbers, for both of these definitions are impredicative.
Gist of Idea
The usual definitions of identity and of natural numbers are impredicative
Source
David Bostock (Philosophy of Mathematics [2009], 8.3)
Book Ref
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.246
A Reaction
[Bostock 237-8 gives details]
Related Idea
Idea 14423
'0', 'number' and 'successor' cannot be defined by Peano's axioms [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]
|
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]
|
18133
|
The usual definitions of identity and of natural numbers are impredicative
[Bostock]
|
18132
|
The predicativity restriction makes a difference with the real numbers
[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]
|