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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I

[axiom concerning what makes a set]

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