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Single Idea 17902
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
]
Full Idea
For any set x, we define the 'successor' of x to be the set S(x) = x U {x}.
Gist of Idea
A successor is the union of a set with its singleton
Source
A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3)
Book Ref
George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.58
A Reaction
This is the Fregean approach to successor, where the Dedekind approach takes 'successor' to be a primitive. Frege 1884:§76.
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]
|
10625
|
Frege had a motive to treat numbers as objects, but not a justification
[Hale/Wright on Frege]
|
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]
|
17636
|
A cardinal number may be defined as a class of similar classes
[Frege, by Russell]
|
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]
|
17820
|
If you can subdivide objects many ways for counting, you can do that to set-elements too
[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]
|