Single Idea 9900

[catalogued under 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 Reference

'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]