MHB What are the key differences between a set and a class?

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A set can be considered a type of class, but not all classes qualify as sets. Sets can contain other sets, while classes cannot contain other classes in the same way. Classes are broader collections that can lead to contradictions in naive set theory, such as Russell's paradox. Modern set theory, particularly ZFC, employs axioms to define valid sets and operations. A class that cannot be constructed as a set is termed a proper class, exemplified by the class of all sets, which cannot exist as a set.
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What the differences between set and class?
 
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highmath said:
What the differences between set and class?
Hi highmath,

Each set can be viewed as a class, but the converse is not true. You can have sets of sets, but not necessarily sets of classes.

A class is basically any collection you can think of; this was the approach used by Cantor in his initial theory (the naive set theory). However, it soon became apparent that attempting to build a set theory on that basis led to contradictions; one of these contradictions is Russel's paradox.

To solve that problem, modern set theory uses an axiomatic method: there is a set of axioms (the standard system is called ZFC) that define a limited set of operations that can be used to construct sets. The collection of objects that satisfy a property is a class, but, unless you can construct it using the axioms of the theory, you cannot assume that it is a set; in particular, you cannot use it as a member of a set.

A class that is not a set is called a proper class. For example, the class of all sets is a proper class: there is no set of all sets.
 

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