more from this thinker | more from this text
Full Idea
A partial ordering is a 'total ordering' just in case any two members of its field are comparable, that is, either a is R to b, or b is R to a, or a is b.
Gist of Idea
A partial ordering becomes 'total' if any two members of its field are comparable
Source
William D. Hart (The Evolution of Logic [2010], 1)
Book Ref
Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.23
A Reaction
See Idea 13457 for 'partial ordering'. The three conditions are known as the 'trichotomy' condition.
Related Idea
Idea 13457 A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [Hart,WD]
| 14126 | Order rests on 'between' and 'separation' [Russell] |
| 14127 | Order depends on transitive asymmetrical relations [Russell] |
| 11222 | The ordered pair <x,y> is defined as the set {{x},{x,y}}, capturing function, not meaning [Gupta] |
| 13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD] |
| 13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [Hart,WD] |
| 13458 | A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD] |
| 13490 | Von Neumann defines α<β as α∈β [Hart,WD] |
| 13653 | 'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro] |
| 15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
| 17759 | Ordinals play the central role in set theory, providing the model of well-ordering [Walicki] |