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Single Idea 10861

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers ]

Full Idea

With ordinary finite numbers ordinals and cardinals are in effect the same, but beyond infinity it is possible for two sets to have the same cardinality but different ordinals.

Gist of Idea

Beyond infinity cardinals and ordinals can come apart

Source

Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13)

Book Ref

Clegg,Brian: 'Infinity' [Robinson 2003], p.169

The 20 ideas from 'Infinity: Quest to Think the Unthinkable'

10854 Transcendental numbers can't be fitted to finite equations [Clegg]
10853 Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg]
10858 By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg]
10861 Beyond infinity cardinals and ordinals can come apart [Clegg]
10860 An ordinal number is defined by the set that comes before it [Clegg]
10859 A set is 'well-ordered' if every subset has a first element [Clegg]
10857 Set theory made a closer study of infinity possible [Clegg]
10864 Any set can always generate a larger set - its powerset, of subsets [Clegg]
10862 The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg]
10866 Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg]
10869 The Continuum Hypothesis is independent of the axioms of set theory [Clegg]
10872 Extensionality: Two sets are equal if and only if they have the same elements [Clegg]
10874 Specification: a condition applied to a set will always produce a new set [Clegg]
10877 Powers: All the subsets of a given set form their own new powerset [Clegg]
10879 Choice: For every set a mechanism will choose one member of any non-empty subset [Clegg]
10871 Axiom of Existence: there exists at least one set [Clegg]
10878 Infinity: There exists a set of the empty set and the successor of each element [Clegg]
10875 Pairing: For any two sets there exists a set to which they both belong [Clegg]
10876 Unions: There is a set of all the elements which belong to at least one set in a collection [Clegg]
10880 Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg]