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Unlike axiomatic systems deal with classes such as NBG, the term "class" is not a word in ZFC. How do I formally treat classes?

Here is an example of what I'm exactly talking about.

Example

Consider a polynomial ring $R[X]$ over a ring $R$.

We call $X$ as the "indeterminate" and write elements of $R[X]$ as $a_0+a_1X+\cdots a_n X^n$.

This is just a convention. This notation DOES NOT mean that it is a sum of $a_kX^k$, since one has NOT defined $X$.

However, $X$ can be formally defined if one defines $R[X]$ as the set of functions $\omega\rightarrow R$ with a finite support and define $X$ as $f(i)=\delta_{i1}$.

Just like this example, I want to know how to formally treat "class".

Here is a phrase in Jech-Set theory

p. 3

Although we work in ZFC which, unlike alternative axiomatic set theories, has one type of object, namely sets, we introduce the informal notion of a class.

If $\phi(x,p_1,\cdots,p_n)$ is a formula, we call $\mathbb{C}=\{x:\phi(x,p_1,\cdots,p_n)\}$ a class.

To be very precise, $\mathbb{C}$ defined here is just a drawing, not a mathematical object.

Here is an example to make clear what I'm trying to ask:

Let's assume there is a guy who only knows English.

To him, only meaningful combinations of alphabets are the meaningful words. The combination such as "adssyfeq" is not a word to him. (Hence, we can consider "he is a mathematician using sets under ZFC")

One day, he faces a combination of Chinese alphabets. Well, to him, this combination of letters is just a drawing and there is no way to make this word meaningful unless he learns Chinese which is a completely different from English.

Just like this example, is it legit to use metalanguage such as $\{x:x=x\}$ in ZFC? Is $\{x:x=x\}$ a object we can talk about in ZFC?

Also, I heard that one can treat and define classes formally if one allows inaccessible cardinal axiom. How?

Thank you in advance..

Community
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Rubertos
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  • It's easy enough to translate statements about classes into valid statements of ZFC. Say that $\mathscr{C}={x;\vert;\phi(x)}$ is a class. Then the statement "$\psi(x)$ holds for all $x\in\mathscr{C}$ is really just a way of saying $\forall x,(\phi(x)\to\psi(x))$. But $\mathscr{C}$ itself doesn't really exist. ${x;\vert;x=x}$ doesn't exist in ZFC (for if it did then ZFC would be inconsistent). – Marc Oct 27 '14 at 05:16
  • @Marc So do you mean the notion of "class" is just a convention in this case? (Jech's informal definition) – Rubertos Oct 27 '14 at 05:18
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    Yes, speaking of classes is just a convention. In Jech's book, any class $\mathbf{C}$ is defined by some formula with parameters $\varphi(x,p_1,\dots,p_n)$. Any statement about $\mathbf{C}$ is really just a more intuitive way of abbreviating a statement involving the formula $\varphi$ that never mentions classes. You just need to replace statements like $x \in \mathbf{C}$, $\mathbf{C} = \mathbf{D}$, $\mathbf{C} \subseteq \mathbf{D}$, etc., with the corresponding statements concerning the formulas defining $\mathbf{C}$ and $\mathbf{D}$. – user187373 Oct 27 '14 at 05:50
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    The first rule of ZFC club is we don't talk about classes ... – Hagen von Eitzen Oct 18 '15 at 12:08

1 Answers1

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You can formally define classes in the meta-theory.

If the meta-theory is a set theory, then you can think about the universe $V$ of sets as just a set model in the meta-theory's universe. In that case a class is just a definable (with parameters) subset of $V$.

If the meta-theory is some arithmetic theory, or otherwise incapable of expressing semantics as sets (meaning that we work syntactically with $\sf ZFC$ and proofs, rather than taking a model of set theory and so on), the a class is really just a formula which has possible parameters, and we say things like "The relativization of the axioms of $\sf ZFC$ to the class defined by this formula is provable from the axioms of $\sf ZFC$" (with the additional quantifiers about parameters when needed, e.g. there is a choice of parameters such that ... or for every choice of parameters ...)

Assuming an inaccessible cardinal makes things easy because it gives us a set universe which agrees with the meta-theory's universe about the power set operation, something which is not to be underestimated. But of course we can do with much much much less than that. Any set model of $\sf ZFC$ will do, it doesn't even have to be a well-founded one.

Asaf Karagila
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  • Would you please recommend me series of books to learn foundations? I want to start from the very basepoint. I think Jech's set theory is not the first one in the hierarchy(?) to learn foundation. I'm asking this since now I think I should learn meta-theory first before I learn serious set theory. I'm even thinking of majoring foundation. – Rubertos Oct 27 '14 at 05:40
  • What does it mean "learn meta-theory first"? – Asaf Karagila Oct 27 '14 at 05:44
  • Umm.. I have learned "set theory" when I was sophomore. Now I'm planning to study foundation seriously. Since you are majoring foundation, I'm sure you know that if you learn foundation once again, then study what first then study what next. (If you want, at the end, to master it) For example, if I have to learn undergraduate mathematics again in next life then I would learn topology and abstract algebra first then study analysis and differential equations and others. – Rubertos Oct 27 '14 at 05:54
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    This is an issue of crutches and scaffolds. Studying foundations without already having a good grasp on what you're trying to build on that foundation is not a good idea. The correct way, as I see it, is to study the basics of first-order logic first, then study both set theory and more advance topics about logic. This will end up with having a good firm grasp of set theory, as well basics of logic. Then things start resolving themselves, and you just understand that you already know the answer to your questions. – Asaf Karagila Oct 27 '14 at 06:21
  • I see. Since there's no prof majoring foundation in my university, I want to get a recommendation from someone who majors it. Urr.. Would you recommend me a first-order logic text? To be honest I have studied (not seriously It took like some days) some first-order logic texts but I'm extremely not satisfied with those. – Rubertos Oct 27 '14 at 06:27
  • I haven't read any intro books, but I think that Enderton's book is pretty solid. http://math.stackexchange.com/questions/4170/good-books-on-mathematical-logic has more information. You can also find a concise overview of logic in Halbeisen's "Combinatorial Set Theory" book. But I'd recommend that you first have a sure footing in the basics of logic before going there. – Asaf Karagila Oct 27 '14 at 06:35
  • @AsafKaragila Meta-theory? So, classes cannot be formalized using only ZFC. You need some other axioms in addition to ZFC. Likewise for cardinal numbers. Correct? – Dan Christensen Oct 29 '14 at 14:53
  • @Dan: Classes are not objects of the universe of $\sf ZFC$. If you mean something else by "formalize", then you should say so. I'm not sure how you concluded that cardinal numbers can't be formalized in $\sf ZFC$, but if there's one thing that the past discussions with you have taught me, is to try and avoid additional questions. So I'm not going to continue this discussion now. Maybe try and pick up a book instead of misinterpreting people online. – Asaf Karagila Oct 29 '14 at 14:56
  • @AsafKaragila Thanks for confirming that classes cannot be defined in ZFC alone. And that, somehow, cardinal numbers can be. – Dan Christensen Oct 29 '14 at 15:04