I am studying category theory for the first time, and I keep coming across the idea that sets cannot be “too big”.

For example, in the definition of a small category: A category is small if the collection of objects and the collection of arrows are sets.

My question is this; given some collection, how can I systematically decide if it is a set? And if it is not a set, how can I tell how far removed it is from a set? (In other words, what is it?)

I am familiar with Russel’s paradox, but I’m not sure how that is useful to determine whether or not something is a set.

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    $\begingroup$ I think you need ZFC here, where the set is defined as an element of a model satisfying the axioms. May be it's not the case when speaking about categories. P.S. The collections that are not sets are called proper classes. $\endgroup$ Jan 27 '20 at 0:22
  • $\begingroup$ Thanks, that makes sense. I’m finding the axioms a bit hard to digest on the Wikipedia page though. Do you know of any books that explain ZFC well? $\endgroup$
    – Owen
    Jan 27 '20 at 0:32
  • $\begingroup$ @Owen I think you can read the first two chapters of Mendelson's Introduction to mathematical logic. He explains very clear how theories work. After that it will be just a mechanical procedure to learn ZFC's axioms. $\endgroup$ Jan 27 '20 at 0:39
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    $\begingroup$ Consider the class of all groups. Is the class of all bijections from this class to itself a group? If so, it would be a member of the class of all groups. Is there something wrong with that? $\endgroup$ Jan 27 '20 at 2:44

Math Stack Exchange has some contributors that are capable of giving much better answers to this question than I can give, but I'm going to try to answer the question as best as I can. Maybe some of you can help me improve this response.

There is a list of axioms for set theory called ZFC, which is short for "Zermelo-Fraenkel Set Theory, including the Axiom of Choice". There are other sets of axioms for set theory, but ZFC is what most mathematicians use. The axioms of ZFC tell you things like:

$$\text{If }X\text{ and }Y\text{ are sets, then so is } \{ X,Y \}.$$

$$\text{If }X\text{ is a set, then so is the union }\bigcup X.$$

$$\text{If }X\text{ is a set, then so is the power set }\mathcal{P}(X).$$

$$\text{If }X\text{ is a set and }\phi\text{ is a formula, then } \{ y\in X\,\vert\,\phi(y) \} \text{ is a set.}$$

Using the axioms of ZFC, you can also prove things like

$$\text{If }X\text{ and }Y\text{ are sets, then so is }X\cap Y.$$

$$\text{If }X\text{ and }Y\text{ are sets, then so is }X\times Y.$$

$$\text{If }X\text{ and }Y\text{ are sets, then the set of functions that map }X\to Y\text{ is really a set.}$$

You can also prove that $\varnothing$ is a set. And using $\varnothing$, and the axioms of ZFC, you can construct $\Bbb{N}$, $\Bbb{Z}$, $\Bbb{Q}$, $\Bbb{R}$, and $\Bbb{C}$ and show that these are all sets. The hard part is constructing $\Bbb{N}$. The basic idea is to let $0=\varnothing$, and let $n=\{0,1,\ldots,n-1\}$. And then, one of the ZFC axioms -- the axiom of infinity -- tells you that there's a set containing $0,1,2,\ldots$.

So basically ZFC tells you that (1) the basic methods for building sets can be used to build new sets out of old ones, and that (2) important familiar things -- such as $\Bbb{N}$, $\Bbb{Z}$, $\Bbb{Q}$, $\Bbb{R}$, and $\Bbb{C}$ -- really are sets.

To get the exact statement of all the axioms of ZFC, you can look at this Wikipedia page

To learn more about how to do the above, or about the subtleties involved, you should have a look at a book about axiomatic set theory. Since you already know some math, I would recommend the book "Foundations of Mathematics" by Kenneth Kunen. Not only is this the best introduction to the subject that I've ever seen, the book only costs $\$26$ on Amazon.com

Addendum: A collection of objects that are defined using some formula is called a class. A class that is not a set is called a proper class. For example, the class of all sets is a proper class. You can prove this using ZFC. The proof is basically to use the idea behind Russell's Paradox: if the class of all sets was a set $X$, then ZFC would tell you that $\{Y\in X\,\vert\,Y\notin Y\}$ would also be a set, which as you know, leads to a contradiction.

  • $\begingroup$ Fantastic, thank you! $\endgroup$
    – Owen
    Jan 27 '20 at 0:58
  • $\begingroup$ No problem. I hope this helped. By the way, Kunen's book really is beautiful. I think it will be worth your time. $\endgroup$
    – user729424
    Jan 27 '20 at 0:59
  • $\begingroup$ @Andrew Ostergaard Should I read the whole book and solve all exercises? I have wanted to study more about set theory and logic and I found the book you recommended interesting. $\endgroup$
    – user682705
    Jan 27 '20 at 9:07
  • $\begingroup$ @user682705 I think whether math people should read Kunen's book depends on how close their interests are to set theory and logic. Kunen's book wouldn't be the most important thing to read if you wanted to be an analytic number theorist. If your interest is in category theory, then it's great to be familiar with the material in Kunen's book, because then you won't have to sweep all these set-theoretic issues under the rug. I loved Kunen's book. Since you said you wanted to learn more about set theory and logic, then I think that's probably a good enough reason to read it. $\endgroup$
    – user729424
    Jan 27 '20 at 13:38
  • $\begingroup$ @Owen: The axioms Andrew mentioned give you ZC (Zermelo set theory plus Choice). ZC is sufficient for most theorems in ordinary mathematics, including the well-ordering theorem and transfinite induction/recursion along any well-ordering. You rarely need to go further unless you are testing the limits of set-theoretic constructions. Even then, you almost always need only Replacement. $\endgroup$
    – user21820
    Feb 2 '20 at 15:45

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