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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
 14126 Order rests on 'between' and 'separation' [Russell]
 14127 Order depends on transitive asymmetrical relations [Russell]
 11222 The ordered pair 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]
 13458 A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD]
 13457 A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [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]