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Single Idea 7738
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
]
Full Idea
Zero is the extension of 'is equinumerous with the concept "is not self-identical"' (which holds of no objects); ..one is defined as the extension of 'is equinumerous with the concept "is identical to zero"'.
Clarification
The 'extension' is the members of that set
Gist of Idea
Zero is defined using 'is not self-identical', and one by using the concept of zero
Source
report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Joan Weiner - Frege Ch.4
Book Ref
Weiner,Joan: 'Frege' [OUP 1999], p.66
A Reaction
It sounds like some sort of cheating to define zero in terms of objects, but one in terms of concepts.
The
46 ideas
with the same theme
[Frege's view of numbers as extensions of classes]:
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]
|
17820
|
If you can subdivide objects many ways for counting, you can do that to set-elements too
[Yourgrau on Frege]
|
10625
|
Frege had a motive to treat numbers as objects, but not a justification
[Hale/Wright on Frege]
|
17636
|
A cardinal number may be defined as a class of similar classes
[Frege, by Russell]
|
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]
|
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]
|
7738
|
Zero is defined using 'is not self-identical', and one by using the concept of zero
[Frege, by Weiner]
|
23456
|
Frege said logical predication implies classes, which are arithmetical objects
[Frege, by Morris,M]
|
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]
|
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]
|