8737 | Kant suggested that arithmetic has no axioms [Kant] |
5557 | Axioms ought to be synthetic a priori propositions [Kant] |
8742 | The only axioms needed are for equality, addition, and successive numbers [Mill] |
13508 | Dedekind gives a base number which isn't a successor, then adds successors and induction [Hart,WD on Dedekind] |
16883 | Arithmetical statements can't be axioms, because they are provable [Frege] |
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' [Blackburn on Peano] |
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] |
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] |
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] |