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Single Idea 10877

[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI ]

Full Idea

Axiom of Powers: For each set there exists a collection of sets that contains amongst its elements all the subsets of the given set.

Gist of Idea

Powers: All the subsets of a given set form their own new powerset

Source

Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15)

Book Ref

Clegg,Brian: 'Infinity' [Robinson 2003], p.206

A Reaction

Obviously this must include the whole of the base set (i.e. not just 'proper' subsets), otherwise the new set would just be a duplicate of the base set.

The 20 ideas from Brian Clegg

10854 Transcendental numbers can't be fitted to finite equations [Clegg]
10853 Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg]
10858 By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg]
10861 Beyond infinity cardinals and ordinals can come apart [Clegg]
10860 An ordinal number is defined by the set that comes before it [Clegg]
10859 A set is 'well-ordered' if every subset has a first element [Clegg]
10857 Set theory made a closer study of infinity possible [Clegg]
10864 Any set can always generate a larger set - its powerset, of subsets [Clegg]
10862 The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg]
10866 Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg]
10869 The Continuum Hypothesis is independent of the axioms of set theory [Clegg]
10872 Extensionality: Two sets are equal if and only if they have the same elements [Clegg]
10874 Specification: a condition applied to a set will always produce a new set [Clegg]
10877 Powers: All the subsets of a given set form their own new powerset [Clegg]
10879 Choice: For every set a mechanism will choose one member of any non-empty subset [Clegg]
10871 Axiom of Existence: there exists at least one set [Clegg]
10878 Infinity: There exists a set of the empty set and the successor of each element [Clegg]
10875 Pairing: For any two sets there exists a set to which they both belong [Clegg]
10876 Unions: There is a set of all the elements which belong to at least one set in a collection [Clegg]
10880 Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg]