more from this thinker     |     more from this text

Single Idea 18134

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

Full Idea

Classical mathematicians say predicative mathematics omits areas of great interest, all concerning non-denumerable real numbers, such as claims about huge cardinals. There cannot be a predicative version of this theory.

Gist of Idea

Predicativism makes theories of huge cardinals impossible

Source

David Bostock (Philosophy of Mathematics [2009], 8.3)

Book Ref

Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.251

A Reaction

I'm not sure that anyone will really miss huge cardinals if they are prohibited, though cryptography seems to flirt with such things. Are we ever allowed to say that some entity conjured up by mathematicians is actually impossible?

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]