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### 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle

#### [view that one-one correspondence is basis of numbers]

19 ideas
 8649 Two numbers are equal if all of their units correspond to one another [Hume]
 9956 'The number of Fs' is the extension (a collection of first-level concepts) of the concept 'equinumerous with F' [Frege, by George/Velleman]
 13527 Frege's cardinals (equivalences of one-one correspondences) is not permissible in ZFC [Frege, by Wolf,RS]
 22292 Hume's Principle fails to implicitly define numbers, because of the Julius Caesar [Frege, by Potter]
 17442 Frege thinks number is fundamentally bound up with one-one correspondence [Frege, by Heck]
 14425 A number is something which characterises collections of the same size [Russell]
 18145 Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock]
 18149 There are many criteria for the identity of numbers [Bostock]
 18148 Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock]
 10140 We derive Hume's Law from Law V, then discard the latter in deriving arithmetic [Wright,C, by Fine,K]
 8692 Frege has a good system if his 'number principle' replaces his basic law V [Wright,C, by Friend]
 17440 Wright says Hume's Principle is analytic of cardinal numbers, like a definition [Wright,C, by Heck]
 13893 It is 1-1 correlation of concepts, and not progression, which distinguishes natural number [Wright,C]
 8784 Neo-logicism founds arithmetic on Hume's Principle along with second-order logic [Hale/Wright]
 10529 If Hume's Principle can define numbers, we needn't worry about its truth [Fine,K]
 10530 Hume's Principle is either adequate for number but fails to define properly, or vice versa [Fine,K]
 8266 Simple counting is more basic than spotting that one-to-one correlation makes sets equinumerous [Lowe]
 8302 Fs and Gs are identical in number if they one-to-one correlate with one another [Lowe]
 10133 Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman]