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4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets

[ordered sets, and using sets to describe orderings]

10 ideas
Order rests on 'between' and 'separation' [Russell]
Order depends on transitive asymmetrical relations [Russell]
The ordered pair <x,y> is defined as the set {{x},{x,y}}, capturing function, not meaning [Gupta]
A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD]
A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [Hart,WD]
'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD]
Von Neumann defines α<β as α∈β [Hart,WD]
'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro]
A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine]
Ordinals play the central role in set theory, providing the model of well-ordering [Walicki]