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Full Idea
The Axiom of Reducibility asserted that to every non-elementary function there is an equivalent elementary function [note: two functions are equivalent when the same arguments render them both true or both false].
Gist of Idea
Reducibility: to every non-elementary function there is an equivalent elementary function
Source
Frank P. Ramsey (The Foundations of Mathematics [1925], §2)
Book Ref
Ramsey,Frank: 'Philosophical Papers', ed/tr. Mellor,D.H. [CUP 1990], p.191
A Reaction
Ramsey in the business of showing that this axiom from Russell and Whitehead is not needed. He says that the axiom seems to be needed for induction and for Dedekind cuts. Since the cuts rest on it, and it is weak, Ramsey says it must go.
| 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] |