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6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers

[general ideas about giving arithmetic a formal basis]

19 ideas
Kant suggested that arithmetic has no axioms [Kant, by Shapiro]
Axioms ought to be synthetic a priori propositions [Kant]
The only axioms needed are for equality, addition, and successive numbers [Mill, by Shapiro]
Dedekind gives a base number which isn't a successor, then adds successors and induction [Dedekind, by Hart,WD]
Arithmetical statements can't be axioms, because they are provable [Frege, by Burge]
If principles are provable, they are theorems; if not, they are axioms [Frege]
Numbers have been defined in terms of 'successors' to the concept of 'zero' [Peano, by Blackburn]
Number theory just needs calculation laws and rules for integers [Hilbert]
The definition of order needs a transitive relation, to leap over infinite intermediate terms [Russell]
Axiom of Archimedes: a finite multiple of a lesser magnitude can always exceed a greater [Russell]
It is conceivable that the axioms of arithmetic or propositional logic might be changed [Putnam]
For Zermelo 3 belongs to 17, but for Von Neumann it does not [Benacerraf]
The successor of x is either x and all its members, or just the unit set of x [Benacerraf]
Mathematics is generalisations about singleton functions [Lewis]
The number of Fs is the 'successor' of the Gs if there is a single F that isn't G [Smith,P]
All numbers are related to zero by the ancestral of the successor relation [Smith,P]
Mereological arithmetic needs infinite objects, and function definitions [Reck/Price]
The truth of the axioms doesn't matter for pure mathematics, but it does for applied [Mares]
It is more explanatory if you show how a number is constructed from basic entities and relations [Koslicki]