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2 ideas
17855 | It may be possible to define induction in terms of the ancestral relation [Frege, by Wright,C] |
Full Idea: Frege's account of the ancestral has made it possible, in effect, to define the natural numbers as entities for which induction holds. | |
From: report of Gottlob Frege (Begriffsschrift [1879]) by Crispin Wright - Frege's Concept of Numbers as Objects 4.xix | |
A reaction: This is the opposite of the approach in the Peano Axioms, where induction is used to define the natural numbers. |
10607 | Frege's logic has a hierarchy of object, property, property-of-property etc. [Frege, by Smith,P] |
Full Idea: Frege's general logical system involves a type hierarchy, distinguishing objects from properties from properties-of-properties etc., with every item belonging to a determinate level. | |
From: report of Gottlob Frege (Begriffsschrift [1879]) by Peter Smith - Intro to Gödel's Theorems 14.1 | |
A reaction: The Theory of Types went on to apply this hierarchy to classes, where Frege's disastrous Basic Law V flattens the hierarchy of classes, putting them on the same level (Smith p.119) |