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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 5 ideas with the same theme [axiom for a new set from any given set]:

13038 Power Set: ∀x ∃y ∀z(z ⊂ x → z ∈ y) [Kunen]
13023 The Power Set Axiom is needed for, and supported by, accounts of the continuum [Maddy]
10877 Powers: All the subsets of a given set form their own new powerset [Clegg]
15936 The Power Set is just the collection of functions from one collection to another [Lavine]
18845 If the totality of sets is not well-defined, there must be doubt about the Power Set Axiom [Rumfitt]