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Full Idea
The Axiom of Reducibility says 'There is a type of a-functions such that, given any a-function, it is formally equivalent to some function of the type in question'. ..It involves all that is really essential in the theory of classes. But is it true?
Clarification
'a-functions' are all the functions which can take object a as an argument
Gist of Idea
Reducibility: a family of functions is equivalent to a single type of function
Source
Bertrand Russell (Introduction to Mathematical Philosophy [1919], XVII)
Book Ref
Russell,Bertrand: 'Introduction to Mathematical Philosophy' [George Allen and Unwin 1975], p.191
A Reaction
I take this to say that in the theory of types, it is possible to reduce each level of type down to one type.
| 14459 | Reducibility: a family of functions is equivalent to a single type of function [Russell] |
| 21720 | Russell saw Reducibility as legitimate for reducing classes to logic [Linsky,B on Russell/Whitehead] |
| 18130 | Axiom of Reducibility: there is always a function of the lowest possible order in a given level [Russell, by Bostock] |
| 13428 | Reducibility: to every non-elementary function there is an equivalent elementary function [Ramsey] |
| 21716 | In simple type theory the axiom of Separation is better than Reducibility [Gödel, by Linsky,B] |
| 18170 | The Axiom of Reducibility is self-effacing: if true, it isn't needed [Quine] |
| 21717 | Reducibility undermines type ramification, and is committed to the existence of functions [Quine, by Linsky,B] |
| 18169 | Axiom of Reducibility: propositional functions are extensionally predicative [Maddy] |
| 21705 | Reducibility says any impredicative function has an appropriate predicative replacement [Linsky,B] |
| 17900 | The Axiom of Reducibility made impredicative definitions possible [George/Velleman] |