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Single Idea 8672

[filed under theme 4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST ]

Full Idea

The 'powerset' of a set is a set made up of all the subsets of a set. For example, the powerset of {3,7,9} is {null, {3}, {7}, {9}, {3,7}, {3,9}, {7,9}, {3,7,9}}. Taking the powerset of an infinite set gets us from one infinite cardinality to the next.

Gist of Idea

A 'powerset' is all the subsets of a set

Source

Michèle Friend (Introducing the Philosophy of Mathematics [2007], 1.5)

Book Ref

Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.21

A Reaction

Note that the null (empty) set occurs once, but not in the combinations. I begin to have queasy sympathies with the constructivist view of mathematics at this point, since no one has the time, space or energy to 'take' an infinite powerset.

The 40 ideas with the same theme [main concepts which are distinctive of set theory]:

13201 ∈ says the whole set is in the other; ⊆ says the members of the subset are in the other [Enderton]
13204 The 'ordered pair' <x,y> is defined to be {{x}, {x,y}} [Enderton]
13206 A 'linear or total ordering' must be transitive and satisfy trichotomy [Enderton]
9699 The 'powerset' of a set is all the subsets of a given set [Enderton]
9700 Two sets are 'disjoint' iff their intersection is empty [Enderton]
9702 A 'domain' of a relation is the set of members of ordered pairs in the relation [Enderton]
9701 A 'relation' is a set of ordered pairs [Enderton]
9706 A 'function' is a relation in which each object is related to just one other object [Enderton]
9708 A function 'maps A into B' if the relating things are set A, and the things related to are all in B [Enderton]
9709 A function 'maps A onto B' if the relating things are set A, and the things related to are set B [Enderton]
9711 A relation is 'reflexive' on a set if every member bears the relation to itself [Enderton]
9714 A relation satisfies 'trichotomy' if all pairs are either relations, or contain identical objects [Enderton]
9717 A set is 'dominated' by another if a one-to-one function maps the first set into a subset of the second [Enderton]
9712 A relation is 'symmetric' on a set if every ordered pair has the relation in both directions [Enderton]
9713 A relation is 'transitive' if it can be carried over from two ordered pairs to a third [Enderton]
12337 There is 'transivity' iff membership ∈ also means inclusion ⊆ [Badiou]
15500 Classes divide into subclasses in many ways, but into members in only one way [Lewis]
15499 A subclass of a subclass is itself a subclass; a member of a member is not in general a member [Lewis]
18194 'Forcing' can produce new models of ZFC from old models [Maddy]
9695 An 'ordered pair' (or ordered n-tuple) is a set with its members in a particular order [Priest,G]
9696 A 'cartesian product' of sets is the set of all the n-tuples with one member in each of the sets [Priest,G]
9686 A 'set' is a collection of objects [Priest,G]
9687 A 'member' of a set is one of the objects in the set [Priest,G]
9688 A 'singleton' is a set with only one member [Priest,G]
9689 The 'empty set' or 'null set' has no members [Priest,G]
9690 A set is a 'subset' of another set if all of its members are in that set [Priest,G]
9691 A 'proper subset' is smaller than the containing set [Priest,G]
9694 The 'relative complement' is things in the second set not in the first [Priest,G]
9693 The 'intersection' of two sets is a set of the things that are in both sets [Priest,G]
9692 The 'union' of two sets is a set containing all the things in either of the sets [Priest,G]
9698 The 'induction clause' says complex formulas retain the properties of their basic formulas [Priest,G]
10890 A 'partial ordering' is reflexive, antisymmetric and transitive [Zalabardo]
10889 The 'Cartesian Product' of two sets relates them by pairing every element with every element [Zalabardo]
10098 The 'power set' of A is all the subsets of A [George/Velleman]
10099 The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman]
10101 Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B [George/Velleman]
10859 A set is 'well-ordered' if every subset has a first element [Clegg]
15914 An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine]
8665 A 'proper subset' of A contains only members of A, but not all of them [Friend]
8672 A 'powerset' is all the subsets of a set [Friend]