Interpretation
To understand this axiom, note that the clause in parentheses in the symbolic statement above simply states that A and B have precisely the same members. Thus, what the axiom is really saying is that two sets are equal if and only if they have precisely the same members. The essence of this is:
- A set is determined uniquely by its members.
The axiom of extensionality can be used with any statement of the form, where P is any unary predicate that does not mention A, to define a unique set whose members are precisely the sets satisfying the predicate . We can then introduce a new symbol for ; it's in this way that definitions in ordinary mathematics ultimately work when their statements are reduced to purely set-theoretic terms.
The axiom of extensionality is generally uncontroversial in set-theoretical foundations of mathematics, and it or an equivalent appears in just about any alternative axiomatisation of set theory. However, it may require modifications for some purposes, as below.
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