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A class is a collection of sets in which the sets are members of the class.

So, may I ask what are the differences between a class and a set (or more precisely a set of sets) by their nature? What motivates mathematicians to give it a separate name?

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J-A-S
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    See Class (set theory) – Mauro ALLEGRANZA Nov 30 '20 at 08:18
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    A property specified by the language is a class. A class that belongs to another class is a set. Motivation: paradoxes of early set theory – Mauro ALLEGRANZA Nov 30 '20 at 08:19
  • This is best answered by logicians or set theorists who have the greatest overview of the different notions of sets and classes. You may also find an answer here (and the question may be a duplicate). In short: Some collections that we want to consider are, in a certain sense, too large to be considered sets, see Russell’s paradox. For these, the concept of classes are introduced. Every set, however, can be considered a class. – k.stm Nov 30 '20 at 08:20
  • Russell’s paradoxon arises when postulating that every collection of which we can make sense, loosely speaking, shall be a set itself. It turns out that in order to resolve this paradoxon, it suffices to postulate that collections are in general something ontologically different, which we then name classes, which we may not combine to new classes so easily and freely as we would like to do with sets, with sets then being only specific such classes of which we then further demand that we may combine them to new sets as we please. – k.stm Nov 30 '20 at 08:32

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  • A set is an element of some class.
  • We typically identify a set with the class that has the same elements, and say that class is a set.
  • A class that isn't a set is not an element of any class or thereby identified with an element of any class, is larger than any set (according to the axiom schema of replacement), and is called a proper class.
  • While arbitrary classes of sets exist, a class is often proven not to be identifiable with any set by showing such a set's existence would imply a contradiction.
  • A theory with only sets has classes in its metalanguage, but discussing proper classes can be convenient. A first-order theory with classes has an in-a-set predicate, typically defined as being an element of a class. There are also second-order theories, where classes don't have the same quantifiers as sets.
J.G.
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