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Full Idea
ZF set theory is seen as a rival to logicism as a foundational scheme. Set theory is for those who have given up the project of reducing mathematics to logic.
Gist of Idea
For those who abandon logicism, standard set theory is a rival option
Source
Bernard Linsky (Russell's Metaphysical Logic [1999], 6.1)
Book Ref
Linsky,Bernard: 'Russell's Metaphysical Logic' [CSLI 1999], p.91
A Reaction
Presumably there are other rivals. Set theory has lots of ontological commitments. One could start at the other end, and investigate the basic ontological commitments of arithmetic. I have no idea what those might be.
| 21704 | 'Impredictative' definitions fix a class in terms of the greater class to which it belongs [Linsky,B] |
| 21705 | Reducibility says any impredicative function has an appropriate predicative replacement [Linsky,B] |
| 21703 | Types are 'ramified' when there are further differences between the type of quantifier and its range [Linsky,B] |
| 21714 | The ramified theory subdivides each type, according to the range of the variables [Linsky,B] |
| 21713 | Did logicism fail, when Russell added three nonlogical axioms, to save mathematics? [Linsky,B] |
| 21715 | For those who abandon logicism, standard set theory is a rival option [Linsky,B] |
| 21719 | Extensionalism means what is true of a function is true of coextensive functions [Linsky,B] |
| 21721 | Higher types are needed to distinguished intensional phenomena which are coextensive [Linsky,B] |
| 21723 | The task of logicism was to define by logic the concepts 'number', 'successor' and '0' [Linsky,B] |
| 21727 | Definite descriptions theory eliminates the King of France, but not the Queen of England [Linsky,B] |
| 21729 | Construct properties as sets of objects, or say an object must be in the set to have the property [Linsky,B] |