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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]
     Full Idea: The two sources of order are 'between' and 'separation'.
     From: Bertrand Russell (The Principles of Mathematics [1903], §204)
Order depends on transitive asymmetrical relations [Russell]
     Full Idea: All order depends upon transitive asymmetrical relations.
     From: Bertrand Russell (The Principles of Mathematics [1903], §208)
The ordered pair <x,y> is defined as the set {{x},{x,y}}, capturing function, not meaning [Gupta]
     Full Idea: The ordered pair <x,y> is defined as the set {{x},{x,y}}. This does captures its essential uses. Pairs <x,y> <u,v> are identical iff x=u and y=v, and the definition satisfies this. Function matters here, not meaning.
     From: Anil Gupta (Definitions [2008], 1.5)
     A reaction: This is offered as an example of Carnap's 'explications', rather than pure definitions. Quine extols it as a philosophical paradigm (1960:§53).
'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD]
     Full Idea: A total order 'well-orders' its field just in case any nonempty subset B of its field has an R-least member, that is, there is a b in B such that for any a in B different from b, b bears R to a. So less-than well-orders natural numbers, but not integers.
     From: William D. Hart (The Evolution of Logic [2010], 1)
     A reaction: The natural numbers have a starting point, but the integers are infinite in both directions. In plain English, an order is 'well-ordered' if there is a starting point.
A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD]
     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.
     From: William D. Hart (The Evolution of Logic [2010], 1)
     A reaction: See Idea 13457 for 'partial ordering'. The three conditions are known as the 'trichotomy' condition.
A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [Hart,WD]
     Full Idea: We say that a binary relation R 'partially orders' a field A just in case R is irreflexive (so that nothing bears R to itself) and transitive. When the set is {a,b}, its subsets {a} and {b} are incomparable in a partial ordering.
     From: William D. Hart (The Evolution of Logic [2010], 1)
Von Neumann defines α<β as α∈β [Hart,WD]
     Full Idea: One of the glories of Von Neumann's theory of numbers is to define α < β to mean that α ∈ β.
     From: William D. Hart (The Evolution of Logic [2010], 3)
'Well-ordering' of a set is an irreflexive, transitive, and binary relation with a least element [Shapiro]
     Full Idea: A 'well-ordering' of a set X is an irreflexive, transitive, and binary relation on X in which every non-empty subset of X has a least element.
     From: Stewart Shapiro (Foundations without Foundationalism [1991], 5.1.3)
     A reaction: So there is a beginning, an ongoing sequence, and no retracing of steps.
A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine]
     Full Idea: A collection M is 'well-ordered' by a relation < if < linearly orders M with a least element, and every subset of M that has an upper bound not in it has an immediate successor.
     From: Shaughan Lavine (Understanding the Infinite [1994], III.4)
Ordinals play the central role in set theory, providing the model of well-ordering [Walicki]
     Full Idea: Ordinals play the central role in set theory, providing the paradigmatic well-orderings.
     From: Michal Walicki (Introduction to Mathematical Logic [2012], 2.3)
     A reaction: When you draw the big V of the iterative hierarchy of sets (built from successive power sets), the ordinals are marked as a single line up the middle, one ordinal for each level.