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Single Idea 10872

[filed under theme 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I ]

Full Idea

Axiom of Extension: Two sets are equal if and only if they have the same elements.

Gist of Idea

Extensionality: Two sets are equal if and only if they have the same elements

Source

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

Book Ref

Clegg,Brian: 'Infinity' [Robinson 2003], p.205

The 9 ideas with the same theme [axiom concerning what makes a set]:

14428 Members define a unique class, whereas defining characteristics are numerous [Russell]
16449 In modal set theory, sets only exist in a possible world if that world contains all of its members [Stalnaker]
13030 Extensionality: ∀x ∀y (∀z (z ∈ x ↔ z ∈ y) → x = y) [Kunen]
13013 The Axiom of Extensionality seems to be analytic [Maddy]
13014 Extensional sets are clearer, simpler, unique and expressive [Maddy]
10097 Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman]
10872 Extensionality: Two sets are equal if and only if they have the same elements [Clegg]
18836 A set may well not consist of its members; the empty set, for example, is a problem [Rumfitt]
18837 A set can be determinate, because of its concept, and still have vague membership [Rumfitt]