axiom of extensionality
Quick Summary
The axiom of extensionality is a fundamental principle in set theory. It states that two sets are considered identical if and only if they contain precisely the same members. In simpler terms, a set is defined by its elements, not by how those elements are described or ordered. This means that sets like {2, 3, 5, 7} and {prime numbers less than 10} are the same set because they have the identical collection of members. This axiom is a cornerstone of the Zermelo-Fraenkel (ZF) set theory, which provides a formal foundation for mathematics.123
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formal logicSet theory…principle is known as the principle of extensionality. A class with no members, such as the class of atheistic popes, is said to be null. Since the membership of all such classes is the same, there is only one null class, which is therefore usually called the null class (or…
Read More…principle is known as the principle of extensionality. A class with no members, such as the class of atheistic popes, is said to be null. Since the membership of all such classes is the same, there is only one null class, which is therefore usually called the null class (or…
Read More- set theoryEssential features of Cantorian set theory
…by what is called the principle of extension—a set is determined by its members rather than by any particular way of describing the set. Thus, sets A and B are equal if and only if every element in A is also in B and every element in B is in…
Read More…by what is called the principle of extension—a set is determined by its members rather than by any particular way of describing the set. Thus, sets A and B are equal if and only if every element in A is also in B and every element in B is in…
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history of logicZermelo-Fraenkel set theory (ZF)…several much more restrictive axioms:
Read More…several much more restrictive axioms:
Read More- set theorySchemas for generating well-formed formulas
The ZFC “axiom of extension” conveys the idea that, as in naive set theory, a set is determined solely by its members. It should be noted that this is not merely a logically necessary property of equality but an assumption about the membership relation as well.
Read MoreThe ZFC “axiom of extension” conveys the idea that, as in naive set theory, a set is determined solely by its members. It should be noted that this is not merely a logically necessary property of equality but an assumption about the membership relation as well.
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