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Full Idea
The Axiom of Extensionality says that for all sets x and y, if x and y have the same elements then x = y.
Gist of Idea
Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = 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.50
A Reaction
This seems fine in pure set theory, but hits the problem of renates and cordates in the real world. The elements coincide, but the axiom can't tell you why they coincide.
| 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] |