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Single Idea 14431
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
]
Full Idea
Order must be defined by means of a transitive relation, since only such a relation is able to leap over an infinite number of intermediate terms. ...Without it we would not be able to define the order of magnitude among fractions.
Gist of Idea
The definition of order needs a transitive relation, to leap over infinite intermediate terms
Source
Bertrand Russell (Introduction to Mathematical Philosophy [1919], IV)
Book Ref
Russell,Bertrand: 'Introduction to Mathematical Philosophy' [George Allen and Unwin 1975], p.38
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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