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6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / c. Fregean numbers

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

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