I'm having a bit of trouble with this one question (3A.2) in Bruce A. Magurn's An Algebraic Introduction to K-Theory. It's about set theory, which has always managed to confuse me more than it should.
Specifically, we are to prove that if $R$ is a ring and 0 is the zero module of $R$, then the isomorphism class cl(0) cannot be a set. The hint that Magurn provides is that if it were a set, then:
(1.) We should be able to construct an argument from the Axiom of Replacement to say that there exists a set $T$ of all sets having only one member.
(2.) From this it would follow that the union of all elements in $T$ is also a set, but this would be a contradiction, since $T$ then would be the set of all sets.
I think that I understand the second part. If $V$ is any given set, then $\{ V \}$ is a set with only one member, and so, since $\{ V \} \in T$, then $V \in \bigcup_T$.
It is the first part that I just cannot work around. How should one construct such an argument?
Any clever person out there who could help me?
The point is that any singleton can be the zero module over $R$. But being a module is more than just being a set, you also come with the necessary operations and actions by the members of $R$.
So a zero module is in fact $(\{0\},+_0,r)_{r\in R}$ tuple. But now using Replacement, we can simply map it to $\{0\}$.
Taking a different set, $x$, we can again define a zero module by using $\{x\}$ as the underlying set, etc.
But the point here is that if the collection of all $0$ modules was a set, then mapping a zero module to its underlying set is a definable set operation, and so by Replacement it will produce a set. But since any singleton can be such module, we would get a set which contains exactly all of the singletons. And you followed the rest quite well.