more from this thinker | more from this text
Full Idea
The Axiom of Reducibility is self-effacing: if it is true, the ramification it is meant to cope with was pointless to begin with.
Gist of Idea
The Axiom of Reducibility is self-effacing: if true, it isn't needed
Source
Willard Quine (Introduction to Russell's Theory of Types [1967], p.152), quoted by Penelope Maddy - Naturalism in Mathematics I.1
Book Ref
Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.12
A Reaction
Maddy says the rejection of Reducibility collapsed the ramified theory of types into the simple theory.
| 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] |