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Single Idea 9191

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers ]

Full Idea

It can be argued that the notion of ordinal numbers is more fundamental than that of cardinals. To count objects, we must count them in sequence. ..The theory of ordinals forms the substratum of Cantor's theory of cardinals.

Gist of Idea

Ordinals seem more basic than cardinals, since we count objects in sequence

Source

Michael Dummett (The Philosophy of Mathematics [1998], 5)

Book Ref

'Philosophy 2: further through the subject', ed/tr. Grayling,A.C. [OUP 1998], p.156

A Reaction

Depends what you mean by 'fundamental'. I would take cardinality to be psychologically prior ('that is a lot of sheep'). You can't order people by height without first acquiring some people with differing heights. I vote for cardinals.

The 19 ideas with the same theme [which type of numbers is the most fundamental?]:

11044 One is prior to two, because its existence is implied by two [Aristotle]
10091 God made the integers, all the rest is the work of man [Kronecker]
10090 Dedekind defined the integers, rationals and reals in terms of just the natural numbers [Dedekind, by George/Velleman]
17452 Ordinals can define cardinals, as the smallest ordinal that maps the set [Dedekind, by Heck]
7524 Order, not quantity, is central to defining numbers [Dedekind, by Monk]
9983 Cantor took the ordinal numbers to be primary [Cantor, by Tait]
18256 Quantity is inconceivable without the idea of addition [Frege]
13510 Could a number just be something which occurs in a progression? [Russell, by Hart,WD]
14128 Some claim priority for the ordinals over cardinals, but there is no logical priority between them [Russell]
14129 Ordinals presuppose two relations, where cardinals only presuppose one [Russell]
14132 Properties of numbers don't rely on progressions, so cardinals may be more basic [Russell]
13489 Von Neumann treated cardinals as a special sort of ordinal [Neumann, by Hart,WD]
18255 Addition of quantities is prior to ordering, as shown in cyclic domains like angles [Dummett]
9191 Ordinals seem more basic than cardinals, since we count objects in sequence [Dummett]
13411 If numbers are basically the cardinals (Frege-Russell view) you could know some numbers in isolation [Benacerraf]
9151 Benacerraf says numbers are defined by their natural ordering [Benacerraf, by Fine,K]
18102 A cardinal is the earliest ordinal that has that number of predecessors [Bostock]
13892 One could grasp numbers, and name sizes with them, without grasping ordering [Wright,C]
8661 The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend]