I am an undergraduate maths student taking a category theory course this term. Very quickly I have run into the issue of how categories such as **Set** can be defined since we can’t have a set of all sets. The lecturer defined categories using the word ‘collection’ but I’ve never encountered a collection in maths that wasn’t a set yet. I’ve seen references to classes when I look this up, which seem to me to be sets but with a different name so that it’s not technically a set of all sets, but then what happens if you need a class of all classes?
The lecturer told us not to worry about it and that he would tell us when we need to consider this issue in our study of category theory but I’m struggling to move on in the class without finding out. I have tried to research this issue but I am struggling to find an explanation at my level.
Please could someone ELI5?
In: 13
You CAN define a set as the set of all sets. There is no problem there.
The set is a member of itself, which is not a problem.
Any set is contained in the set of all sets. No issue.
The set has infinitely many members, but this is also not a problem.
Maybe you’re thinking of the paradox:
“The set of all sets that do not contain themselves.”
This is indeed a paradox. We could go into that if you want (if it’s not self-evident), but to answer your question here: the set of all sets is a perfectly valid set.
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