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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 20 ideas with the same theme [general ideas about giving arithmetic a formal basis]:

23026 We know mathematical axioms, such as subtracting equals from equals leaves equals, by a natural light [Leibniz]
8737 Kant suggested that arithmetic has no axioms [Kant, by Shapiro]
5557 Axioms ought to be synthetic a priori propositions [Kant]
8742 The only axioms needed are for equality, addition, and successive numbers [Mill, by Shapiro]
13508 Dedekind gives a base number which isn't a successor, then adds successors and induction [Dedekind, by Hart,WD]
16883 Arithmetical statements can't be axioms, because they are provable [Frege, by Burge]
16864 If principles are provable, they are theorems; if not, they are axioms [Frege]
3338 Numbers have been defined in terms of 'successors' to the concept of 'zero' [Peano, by Blackburn]
17964 Number theory just needs calculation laws and rules for integers [Hilbert]
14431 The definition of order needs a transitive relation, to leap over infinite intermediate terms [Russell]
14124 Axiom of Archimedes: a finite multiple of a lesser magnitude can always exceed a greater [Russell]
9939 It is conceivable that the axioms of arithmetic or propositional logic might be changed [Putnam]
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]
10808 Mathematics is generalisations about singleton functions [Lewis]
10608 The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P]
10618 All numbers are related to zero by the ancestral of the successor relation [Smith,P]
10174 Mereological arithmetic needs infinite objects, and function definitions [Reck/Price]
17715 The truth of the axioms doesn't matter for pure mathematics, but it does for applied [Mares]
17312 It is more explanatory if you show how a number is constructed from basic entities and relations [Koslicki]