# Classes in Set Theory

Im having a problem with the first rule for class usage enunciated in a book on set theory. ("Basic Set Theory" by Azriel Levy)

T Its the one where a predicate infers some other predicate. It is not clear to me whether A is a set, a class element or a set element and the same for x. The phi's are predicates. I think what is intended to say is that, if a predicate is valid for some element in a set, then a classe can be constructed by all objects satisfying that same predicate Thank you in advance.

• You have not identified the book you quoted from. That is wrong. By the way, what is your question? What do you feel should be more explicit? What is meant by what? Could you be more explicit about what it is you are asking?
– bof
Nov 16, 2018 at 20:45
• That is precisely my point. Before this statement, they talk about usage and rules for classes informally. This os where supposably they start presenting it all more formally. It seemed to me exactly that there were missing some quantifiers or something Nov 16, 2018 at 21:06
• What book did you copy that stuff from?
– bof
Nov 16, 2018 at 23:59
• "Basic Set Theory" by Azriel Levy Nov 17, 2018 at 1:57

## 1 Answer

From pg 11 of the book: "we shall use upper case Roman letters for class variables". Likewise, set variables are lower-case.

The meaning of the inference rule is summarized right under 4.2 as: "whatever holds for all classes holds for all sets." $$\Phi$$ is just some formula, and this has nothing to do with comprehensions. We are interpreting a formula as a universal statement. $$\Phi(A)$$ means "$$\Phi(A)$$ holds for all classes $$A$$" and from that we can infer $$\Phi(x),$$ which means "$$\Phi(x)$$ holds for all sets $$x$$".

• But how could i have know that it could be interpreted in that form if no quantifiers are present? And how could it be quantified over classes? This book says (a little earlier than this stage) that classes can't be quantified over. Nov 17, 2018 at 2:00
• @Daàvid From page 5 "Whenever we use a formula with free variables as an axiom or as a theorem, we mean to say that the formula holds for all possible values given to its free variables." From just under the chapter 4 heading "we shall... introduce a formal system in which class terms and class variables are expressions in their own right and not abbreviations of other expressions. ... We shall now first introduce the formal system for the extended language, and then dwell on the relationship to the basic language." Nov 17, 2018 at 2:13
• @Daàvid I'm not super familiar with this book, but I believe the extended system is something like NBG, where classes can be quantified over, but the subset comprehension axiom is restricted to formulas without class quantifiers. Nov 17, 2018 at 2:21
• Yes, you're right. I understand now why quantifiers weren't used but it still seems to contradict his statement on class's quantification. But it does make sense. Thank you Nov 17, 2018 at 2:55
• @Daàvid Yes, reading more closely, Levy does not allow quantification over class variables, even in the class-set language. Nonetheless, we do have bona fide class variables, and the semantics of open formulas are universal. Note that on the bottom of pg 14, he even says "the extended language does not admit (existential) quantifiers over class variables", alluding to the the fact that some implicit universal quantification over classes has crept in. Nov 17, 2018 at 3:18