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

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SUMMARY

The key differences between a set and a class are foundational in modern set theory. A set can be viewed as a class, but a class does not necessarily qualify as a set. Sets can contain other sets, while classes may include collections that cannot be constructed as sets due to axiomatic limitations. The standard axiomatic system used to define sets is Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), which addresses contradictions such as Russell's paradox by distinguishing between sets and proper classes.

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  • Understanding of Zermelo-Fraenkel set theory (ZFC)
  • Familiarity with the concept of Russell's paradox
  • Basic knowledge of mathematical logic and axiomatic systems
  • Awareness of naive set theory principles
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  • Study the axioms of Zermelo-Fraenkel set theory (ZFC)
  • Explore the implications of Russell's paradox in set theory
  • Learn about proper classes and their role in modern mathematics
  • Investigate the differences between naive set theory and axiomatic set theory
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Mathematicians, logicians, and students of advanced mathematics seeking to deepen their understanding of set theory and its foundational concepts.

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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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