more from this thinker     |     more from this text

Single Idea 10133

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle ]

Full Idea

The derivability of Peano's Postulates from Hume's Principle in second-order logic has been dubbed 'Frege's Theorem', (though Frege would not have been interested, because he didn't think Hume's Principle gave an adequate definition of numebrs).

Gist of Idea

Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle

Source

A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.8 n1)

Book Ref

George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.215

A Reaction

Frege said the numbers were the sets which were the extensions of the sets created by Hume's Principle.

The 19 ideas with the same theme [view that one-one correspondence is basis of numbers]:

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]
18148 Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock]
18149 There are many criteria for the identity of 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]