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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?

StormyTeacup
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1 Answers1

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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.

Asaf Karagila
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  • Why do you not mention that in the standard encodings of ordered pairs or sequences, we do not need Replacement to extract the set of items? Just Union and Specification would do? – user21820 Dec 03 '21 at 06:31
  • Because I don't know how exactly the whole thing was presented. Also, while I love the standard encoding, I despise how non logicians cling to it, as if this is the only way to do it. – Asaf Karagila Dec 03 '21 at 08:37
  • Haha okay. Personally, I would rather have inbuilt pairing constructors and destructors (i.e. function-symbols for them), and pairs would be urelements, thus completely abstracting the pairing notion from the implementation! – user21820 Dec 03 '21 at 08:40
  • Yes, but you're weird. And also not quite a set theorist, from what I remember. – Asaf Karagila Dec 03 '21 at 08:47
  • Definitely not a set theorist, but how's that related? Ironically, in my experience people who bring up the Kuratowski encoding are typically set theorists, since others don't really care. Good to know that I'm not bland. =P – user21820 Dec 03 '21 at 08:49
  • Yes, we bring it up to explain that it can be done. People don't care, that's true, but then they also say that "this is the way to do it" and some even cling to it farther. In any case, if you truly don't care about how the pairs and tuples are coded, then you shouldn't have qualms with invoking Replacement either. :) – Asaf Karagila Dec 03 '21 at 08:57
  • That's a very interesting discussion point. I have no qualms invoking bounded replacement. The funny thing is that we don't need unbounded replacement as long as enough of the things we actually want to do are inbuilt into the language (e.g. pairing, powerset). =) – user21820 Dec 03 '21 at 09:07