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I have two main questions:

1. What is a proper class? I've read that it's collection of objects that's "too big" to be a set, but in what sense is such a collection "too big"? Since I'd like this post to be accessible to people who aren't necessarily with the intricacies of symbolic logic, it'd be much appreciated if any symbolic logic expression be accompanied by an explanation of what it means and maybe a quick example of how it works (e.g. "One consequence of this axiom is that objects such as x={x}, which isn't disjoint from any of it's its elements, can't be a set. However, it [is/is not] a class because...").

2. Beyond the implications for philosophy of mathematics, would most mathematicians ever need to worry about proper classes vs. sets? If so, when and why? If not, what types of mathematicians would need to worry about this distinction?

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# Proper classes and their consequences

I have two main questions:

1. What is a proper class? I've read that it's collection of objects that's "too big" to be a set, but in what sense is such a collection "too big"? Since I'd like this post to be accessible to people who aren't necessarily with the intricacies of symbolic logic, it'd be much appreciated if any symbolic logic expression be accompanied by an explanation of what it means and maybe a quick example of how it works (e.g. "One consequence of this axiom is that objects such as x={x}, which isn't disjoint from any of it's elements, can't be a set. However, it [is/is not] a class because...").

2. Beyond the implications for philosophy of mathematics, would most mathematicians ever need to worry about proper classes vs. sets? If so, when and why? If not, what types of mathematicians would need to worry about this distinction?