In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. It says that sets having the same elements are the same set.
In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:
or in words:
Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B, then A is equal to B.
(It is not really essential that X here be a set — but in ZF, everything is. See Ur-elements below for when this is violated.)
The converse, of this axiom follows from the substitution property of equality.
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.
The axiom given above assumes that equality is a primitive symbol in predicate logic.
Some treatments of axiomatic set theory prefer to do without this, and instead treat the above statement not as an axiom but as a definition of equality.
Then it is necessary to include the usual axioms of equality from predicate logic as axioms about this defined symbol.
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In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908. In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: In words, there is a set I (the set that is postulated to be infinite), such that the empty set is in I, and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I.
In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set. Some mathematicians call it the axiom schema of comprehension, although others use that term for unrestricted comprehension, discussed below.
Zermelo set theory (sometimes denoted by Z-), as set out in a seminal paper in 1908 by Ernst Zermelo, is the ancestor of modern Zermelo–Fraenkel set theory (ZF) and its extensions, such as von Neumann–Bernays–Gödel set theory (NBG). It bears certain differences from its descendants, which are not always understood, and are frequently misquoted. This article sets out the original axioms, with the original text (translated into English) and original numbering.
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