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Single Idea 9904
[filed under theme 6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
]
Full Idea
In no consistent theory is there a class of all classes with seventeen members. The existence of the paradoxes is a good reason to deny to 'seventeen' this univocal role of designating the class of all classes with seventeen members.
Gist of Idea
The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members
Source
Paul Benacerraf (What Numbers Could Not Be [1965], II)
Book Ref
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.284
A Reaction
This was Frege's disaster, and seems to block any attempt to achieve logicism by translating numbers into sets. It now seems unclear whether set theory is logic, or mathematics, or sui generis.
The
24 ideas
from 'What Numbers Could Not Be'
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9151
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Benacerraf says numbers are defined by their natural ordering
[Benacerraf, by Fine,K]
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13891
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To understand finite cardinals, it is necessary and sufficient to understand progressions
[Benacerraf, by Wright,C]
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8697
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Disputes about mathematical objects seem irrelevant, and mathematicians cannot resolve them
[Benacerraf, by Friend]
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8304
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No particular pair of sets can tell us what 'two' is, just by one-to-one correlation
[Benacerraf, by Lowe]
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17904
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A set has k members if it one-one corresponds with the numbers less than or equal to k
[Benacerraf]
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9898
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We can count intransitively (reciting numbers) without understanding transitive counting of items
[Benacerraf]
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17903
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Someone can recite numbers but not know how to count things; but not vice versa
[Benacerraf]
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9897
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The application of a system of numbers is counting and measurement
[Benacerraf]
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17906
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To explain numbers you must also explain cardinality, the counting of things
[Benacerraf]
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9900
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For Zermelo 3 belongs to 17, but for Von Neumann it does not
[Benacerraf]
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9899
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The successor of x is either x and all its members, or just the unit set of x
[Benacerraf]
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9901
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Numbers can't be sets if there is no agreement on which sets they are
[Benacerraf]
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9903
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Number words are not predicates, as they function very differently from adjectives
[Benacerraf]
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9904
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The set-theory paradoxes mean that 17 can't be the class of all classes with 17 members
[Benacerraf]
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9905
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Identity statements make sense only if there are possible individuating conditions
[Benacerraf]
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9906
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If ordinal numbers are 'reducible to' some set-theory, then which is which?
[Benacerraf]
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9907
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If any recursive sequence will explain ordinals, then it seems to be the structure which matters
[Benacerraf]
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9908
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The job is done by the whole system of numbers, so numbers are not objects
[Benacerraf]
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9909
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The number 3 defines the role of being third in a progression
[Benacerraf]
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9911
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Number words no more have referents than do the parts of a ruler
[Benacerraf]
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9910
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Number-as-objects works wholesale, but fails utterly object by object
[Benacerraf]
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9912
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There are no such things as numbers
[Benacerraf]
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8925
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Mathematical objects only have properties relating them to other 'elements' of the same structure
[Benacerraf]
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9938
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How can numbers be objects if order is their only property?
[Benacerraf, by Putnam]
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