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### 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers

#### [Frege's view of numbers as extensions of classes]

45 ideas
 9992 The 'extension of a concept' in general may be quantitatively completely indeterminate [Cantor]
 9949 There is the concept, the object falling under it, and the extension (a set, which is also an object) [Frege, by George/Velleman]
 10623 Frege defined number in terms of extensions of concepts, but needed Basic Law V to explain extensions [Frege, by Hale/Wright]
 9975 Frege ignored Cantor's warning that a cardinal set is not just a concept-extension [Tait on Frege]
 10020 Frege's biggest error is in not accounting for the senses of number terms [Hodes on Frege]
 10553 A number is a class of classes of the same cardinality [Frege, by Dummett]
 9973 The number of F's is the extension of the second level concept 'is equipollent with F' [Frege, by Tait]
 16500 Frege showed that numbers attach to concepts, not to objects [Frege, by Wiggins]
 9990 Frege replaced Cantor's sets as the objects of equinumerosity attributions with concepts [Frege, by Tait]
 17636 A cardinal number may be defined as a class of similar classes [Frege, by Russell]
 7738 Zero is defined using 'is not self-identical', and one by using the concept of zero [Frege, by Weiner]
 13887 Frege started with contextual definition, but then switched to explicit extensional definition [Frege, by Wright,C]
 13897 Each number, except 0, is the number of the concept of all of its predecessors [Frege, by Wright,C]
 9856 Frege's account of cardinals fails in modern set theory, so they are now defined differently [Dummett on Frege]
 9902 Frege's incorrect view is that a number is an equivalence class [Benacerraf on Frege]
 17814 The natural number n is the set of n-membered sets [Frege, by Yourgrau]
 17819 A set doesn't have a fixed number, because the elements can be seen in different ways [Yourgrau on Frege]
 17820 If you can subdivide objects many ways for counting, you can do that to set-elements too [Yourgrau on Frege]
 13871 Frege claims that numbers are objects, as opposed to them being Fregean concepts [Frege, by Wright,C]
 13872 Numbers are second-level, ascribing properties to concepts rather than to objects [Frege, by Wright,C]
 9816 For Frege, successor was a relation, not a function [Frege, by Dummett]
 9953 Numbers are more than just 'second-level concepts', since existence is also one [Frege, by George/Velleman]
 9954 "Number of x's such that ..x.." is a functional expression, yielding a name when completed [Frege, by George/Velleman]
 10139 Frege gives an incoherent account of extensions resulting from abstraction [Fine,K on Frege]
 10028 For Frege the number of F's is a collection of first-level concepts [Frege, by George/Velleman]
 10029 Numbers need to be objects, to define the extension of the concept of each successor to n [Frege, by George/Velleman]
 10625 Frege had a motive to treat numbers as objects, but not a justification [Hale/Wright on Frege]
 16890 Frege's problem is explaining the particularity of numbers by general laws [Frege, by Burge]
 8630 Individual numbers are best derived from the number one, and increase by one [Frege]
 11029 'Exactly ten gallons' may not mean ten things instantiate 'gallon' [Rumfitt on Frege]
 17460 A statement of number contains a predication about a concept [Frege]
 10013 Numerical statements have first-order logical form, so must refer to objects [Frege, by Hodes]
 18181 The Number for F is the extension of 'equal to F' (or maybe just F itself) [Frege]
 18103 Numbers are objects because they partake in identity statements [Frege, by Bostock]
 9586 In a number-statement, something is predicated of a concept [Frege]
 3331 If '5' is the set of all sets with five members, that may be circular, and you can know a priori if the set has content [Benardete,JA on Frege]
 14117 Numbers are properties of classes [Russell]
 17817 Defining 'three' as the principle of collection or property of threes explains set theory definitions [Yourgrau]
 13894 Sameness of number is fundamental, not counting, despite children learning that first [Wright,C]
 12215 The existence of numbers is not a matter of identities, but of constituents of the world [Fine,K]
 18182 The extension of concepts is not important to me [Maddy]
 18177 In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy]
 8297 Numbers are universals, being sets whose instances are sets of appropriate cardinality [Lowe]
 17902 A successor is the union of a set with its singleton [George/Velleman]
 17461 Some 'how many?' answers are not predications of a concept, like 'how many gallons?' [Rumfitt]