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Single Idea 9899
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
]
Full Idea
For Ernie, the successor of a number x was the set consisting of x and all the members of x, while for Johnny the successor of x was simply [x], the unit set of x - the set whose only member is x.
Clarification
Ernie actually refers to Von Neumann's set theory, and Johnny refers to Zermelo's!
Gist of Idea
The successor of x is either x and all its members, or just the unit set of x
Source
Paul Benacerraf (What Numbers Could Not Be [1965], II)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.278
A Reaction
See also Idea 9900. Benacerraf's famous point is that it doesn't seem to make any difference to arithmetic which version of set theory you choose as its basis. I take this to conclusively refute the idea that numbers ARE sets.
Related Ideas
Idea 9900
For Zermelo 3 belongs to 17, but for Von Neumann it does not [Benacerraf]
Idea 8762
Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
The
31 ideas
from Paul Benacerraf
13411
|
If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation
[Benacerraf]
|
13412
|
Obtaining numbers by abstraction is impossible - there are too many; only a rule could give them, in order
[Benacerraf]
|
13413
|
We must explain how we know so many numbers, and recognise ones we haven't met before
[Benacerraf]
|
13415
|
An adequate account of a number must relate it to its series
[Benacerraf]
|
17927
|
Realists have semantics without epistemology, anti-realists epistemology but bad semantics
[Benacerraf, by Colyvan]
|
9935
|
Mathematical truth is always compromising between ordinary language and sensible epistemology
[Benacerraf]
|
9936
|
The platonist view of mathematics doesn't fit our epistemology very well
[Benacerraf]
|
8697
|
Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them
[Benacerraf, by Friend]
|
8304
|
No particular pair of sets can tell us what 'two' is, just by one-to-one correlation
[Benacerraf, by Lowe]
|
9151
|
Benacerraf says numbers are defined by their natural ordering
[Benacerraf, by Fine,K]
|
13891
|
To understand finite cardinals, it is necessary and sufficient to understand progressions
[Benacerraf, by Wright,C]
|
17904
|
A set has k members if it one-one corresponds with the numbers less than or equal to k
[Benacerraf]
|
9898
|
We can count intransitively (reciting numbers) without understanding transitive counting of items
[Benacerraf]
|
17903
|
Someone can recite numbers but not know how to count things; but not vice versa
[Benacerraf]
|
9897
|
The application of a system of numbers is counting and measurement
[Benacerraf]
|
17906
|
To explain numbers you must also explain cardinality, the counting of things
[Benacerraf]
|
9901
|
Numbers can't be sets if there is no agreement on which sets they are
[Benacerraf]
|
9900
|
For Zermelo 3 belongs to 17, but for Von Neumann it does not
[Benacerraf]
|
9899
|
The successor of x is either x and all its members, or just the unit set of x
[Benacerraf]
|
9903
|
Number words are not predicates, as they function very differently from adjectives
[Benacerraf]
|
9904
|
The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members
[Benacerraf]
|
9905
|
Identity statements make sense only if there are possible individuating conditions
[Benacerraf]
|
9906
|
If ordinal numbers are 'reducible to' some set-theory, then which is which?
[Benacerraf]
|
9912
|
There are no such things as numbers
[Benacerraf]
|
9907
|
If any recursive sequence will explain ordinals, then it seems to be the structure which matters
[Benacerraf]
|
9908
|
The job is done by the whole system of numbers, so numbers are not objects
[Benacerraf]
|
9909
|
The number 3 defines the role of being third in a progression
[Benacerraf]
|
9911
|
Number words no more have referents than do the parts of a ruler
[Benacerraf]
|
9910
|
Number-as-objects works wholesale, but fails utterly object by object
[Benacerraf]
|
8925
|
Mathematical objects only have properties relating them to other 'elements' of the same structure
[Benacerraf]
|
9938
|
How can numbers be objects if order is their only property?
[Benacerraf, by Putnam]
|