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Single Idea 10608
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
]
Full Idea
The number of Fs is the 'successor' of the number of Gs if there is an object which is an F, and the remaining things that are F but not identical to the object are equinumerous with the Gs.
Gist of Idea
The number of Fs is the 'successor' of the Gs if there is a single F that isn't G
Source
Peter Smith (Intro to Gödel's Theorems [2007], 14.1)
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.120
The
20 ideas
with the same theme
[general ideas about giving arithmetic a formal basis]:
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23026
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We know mathematical axioms, such as subtracting equals from equals leaves equals, by a natural light
[Leibniz]
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8737
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Kant suggested that arithmetic has no axioms
[Kant, by Shapiro]
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5557
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Axioms ought to be synthetic a priori propositions
[Kant]
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8742
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The only axioms needed are for equality, addition, and successive numbers
[Mill, by Shapiro]
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13508
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Dedekind gives a base number which isn't a successor, then adds successors and induction
[Dedekind, by Hart,WD]
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16883
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Arithmetical statements can't be axioms, because they are provable
[Frege, by Burge]
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16864
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If principles are provable, they are theorems; if not, they are axioms
[Frege]
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3338
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Numbers have been defined in terms of 'successors' to the concept of 'zero'
[Peano, by Blackburn]
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17964
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Number theory just needs calculation laws and rules for integers
[Hilbert]
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14431
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The definition of order needs a transitive relation, to leap over infinite intermediate terms
[Russell]
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14124
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Axiom of Archimedes: a finite multiple of a lesser magnitude can always exceed a greater
[Russell]
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9939
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It is conceivable that the axioms of arithmetic or propositional logic might be changed
[Putnam]
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9900
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For Zermelo 3 belongs to 17, but for Von Neumann it does not
[Benacerraf]
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9899
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The successor of x is either x and all its members, or just the unit set of x
[Benacerraf]
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10808
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Mathematics is generalisations about singleton functions
[Lewis]
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10608
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The number of Fs is the 'successor' of the Gs if there is a single F that isn't G
[Smith,P]
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10618
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All numbers are related to zero by the ancestral of the successor relation
[Smith,P]
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10174
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Mereological arithmetic needs infinite objects, and function definitions
[Reck/Price]
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17715
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The truth of the axioms doesn't matter for pure mathematics, but it does for applied
[Mares]
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17312
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It is more explanatory if you show how a number is constructed from basic entities and relations
[Koslicki]
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