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Hello so currently I am studying Graph Theory and it had a small introduction to sets. I am already familiar with a set but it brought up a new term I never heard before, "class." It says that a class is bigger than a set, so does that mean a set has a limit to the number of characters it has? I have no idea how to wrap my head around this, thanks in advance :)

FShrike
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EtotheEyePie
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    It might help if you provided the context in which "class" came up. Hopefully you are aware that there is a "standard" mathematical foundation, the ZFC set theory. In this foundation, there are difficulties with size - for example, there is no such thing as a "set of all sets" (one way to see this is that for any set $S$, the power set of $S$ always exists (by axiom) has a strictly larger cardinality (Cantor's theorem) hence no set of all sets could contain its own power set). – FShrike Jun 09 '22 at 21:57
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    You might be interested in this post. Classes are sometimes axiomatised in the NBG set theory which is an equiconsistent extension of ZFC (as far as I know, this means we can essentially accept it as safe if we consider ZFC to be safe) – FShrike Jun 09 '22 at 21:57

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The difference between class and set is purely formal. Both are "collections". So to answer this question we need to talk a little about set theories. A set is an object to which the axioms of set theory apply. So a set is an object in ZFC, or some other such theory.

One might wonder about the collection of all sets (in ZFC). But if one assumes that this collection is a set, i.e., that there is a set of all sets, we can generate a paradox by applying the rules of the set theory. Applying the comprehension axiom to the set of all sets can generate Russell's paradox. So, we are forced to say that the collection of all sets is "too big" to be a set because if we treat it like a set we end up in contradiction. So instead, we call it a proper class. So this proper class is like a collection but we cannot apply the rules of set theory to it.

There are other collections which are "too big" to form a set. Examples include the collection of all ordinal numbers, the collection of all groups, ect. This just means that if we apply the standard rules of set theory to these collections, i.e., treat them as sets, then we can generate a paradox.

So what then is a class really? Well, a class can be represented by a formula in first order logic. We think of the class as the collection of all sets which satisfy the formula. But we cannot collect all such sets into one set, because we would then be able to generate a paradox. In most day to day mathematics, you will not need to worry about this distinction.

EDIT. I want to add that there are other things which we do not call sets since the axioms of set theory do not apply to them. An example is the object defined by the formula $x=\{x\}$. We do not call such a thing a set (in ZFC) because it violates one of ZFC's axioms. But we also do not call such a thing a class. Why? Because whatever $x$ is, it does not seem to behave like the intuitive notion of collection. It has an infinite descending chain of elements since we have: $$...\in x\in x\in x$$ We generally think collections are well founded. So this $x$ does not behave, even intuitively, like a collection. So it is not even a collection. Since classes are supposed to be like collections, just too big to apply the rules of set theory, we do not call such a thing a class.

This just illustrates the point that classes are "collection-like", we just cannot formally treat them like sets.

IsAdisplayName
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