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Single Idea 9900

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers ]

Full Idea

Ernie's number progression is [φ],[φ,[φ]],[φ,[φ],[φ,[φ,[φ]]],..., whereas Johnny's is [φ],[[φ]],[[[φ]]],... For Ernie 3 belongs to 17, not for Johnny. For Ernie 17 has 17 members; for Johnny it has one.

Clarification

See also Idea 9899

Gist of Idea

For Zermelo 3 belongs to 17, but for Von Neumann it does not

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

Benacerraf's point is that there is no proof-theoretic way to choose between them, though I am willing to offer my intuition that Ernie (Zermelo) gives the right account. Seventeen pebbles 'contains' three pebbles; you must pass 3 to count to 17.

Related Ideas

Idea 9899 The successor of x is either x and all its members, or just the unit set of x [Benacerraf]

Idea 8762 Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]

The 24 ideas from 'What Numbers Could Not Be'

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]
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]
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]
9899 The successor of x is either x and all its members, or just the unit set of x [Benacerraf]
9900 For Zermelo 3 belongs to 17, but for Von Neumann it does not [Benacerraf]
9901 Numbers can't be sets if there is no agreement on which sets they are [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]
9908 The job is done by the whole system of numbers, so numbers are not objects [Benacerraf]
9907 If any recursive sequence will explain ordinals, then it seems to be the structure which matters [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]
9912 There are no such things as numbers [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]