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Single Idea 14124
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
]
Full Idea
The Axiom of Archimedes asserts that, given any two magnitudes of a kind, some finite multiple of the lesser exceeds the greater.
Gist of Idea
Axiom of Archimedes: a finite multiple of a lesser magnitude can always exceed a greater
Source
Bertrand Russell (The Principles of Mathematics [1903], §168 n*)
Book Ref
Russell,Bertrand: 'Principles of Mathematics' [Routledge 1992], p.181
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]
|
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]
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