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6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / d. Hume's Principle

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

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