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Single Idea 13038

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

Full Idea

Power Set Axiom: ∀x ∃y ∀z(z ⊂ x → z ∈ y). That is, there is a set y which contains all of the subsets of a given set. Hence we define P(x) = {z : z ⊂ x}.

Gist of Idea

Power Set: ∀x ∃y ∀z(z ⊂ x → z ∈ y)

Source

Kenneth Kunen (Set Theory [1980], §1.10)

Book Ref

Kunen,Kenneth: 'Set Theory: Introduction to Independence Proofs' [North-Holland 1980], p.29

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]