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Single Idea 10100

[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II ]

Full Idea

The Axiom of Pairing says that for all sets x and y, there is a set z containing x and y, and nothing else. In symbols: ∀x∀y∃z∀w(w ∈ z ↔ (w = x ∨ w = y)).

Gist of Idea

Axiom of Pairing: for all sets x and y, there is a set z containing just x and y

Source

A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.3)

Book Ref

George,A/Velleman D.J.: 'Philosophies of Mathematics' [Blackwell 2002], p.52

A Reaction

See Idea 10099 for an application of this axiom.

Related Idea

Idea 10099 The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman]

The 5 ideas with the same theme [axiom for new sets by combination]:

13017 Zermelo introduced Pairing in 1930, and it seems fairly obvious [Zermelo, by Maddy]
13032 Pairing: ∀x ∀y ∃z (x ∈ z ∧ y ∈ z) [Kunen]
18851 Pairing (with Extensionality) guarantees an infinity of sets, just from a single element [Rosen]
10100 Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman]
10875 Pairing: For any two sets there exists a set to which they both belong [Clegg]