I'm attempting to teach myself NBG Set Theory from the book Set Theory and the Continuum Problem by Raymond M. Smullyan and Melvin Fitting. Maybe it's just me, but I think the text is confusing. I can get through Principles of Mathematical Analysis by Rudin..., but not this. For example... they do a piss-poor job of defining class and set. As a matter of fact, they don't even offer a formal definition. Here's what the book says: Great. What is a collection? What is a subcollection? What does V look like? And then this: I don't understand this bold part. So a set is a set of sets? Isn't that a class? *sigh*
Both sets and classes are "bags" which hold 0 or more sets. But sets obey more rules than classes. V is the model of the universe of all classes and sets. Its elements are sets and there are a whole lot of them. Sets can have other sets as members without contradiction. In fact there is only one unique set without any members, the empty set. But proper classes are "too big" (i.e. they violate the set axioms) to be members of sets or classes, so we can only describe a proper class as a subclass of the universal class, V. The identity of a set or a class is determined by its contents. A class may be equal in contents to a set which obeys the set axioms, otherwise it is a proper class which does not. // I like set theory as presented by MetaMath. http://metamath.org/ http://us.metamath.org/mpegif/mmset.html#axioms Sets are lowercase red letters, classes (which in this presentation of ZFC set theory are collections of sets that satisfy some logical requirement) are blue purple uppercase letters.
Thank you. I think that I'm now close to an "aha!" moment wherein I can finally move past page 15... Though, I am still having trouble differentiating the difference between classes and sets. The first theorem in the book goes on the prove that not every class is a set... but what good is knowing that if I don't know the difference between a fruit and an apple? Now, I do know that numbers like 1 are actually sets. When we speak of elements in NBG, are we actually speaking of sets?
Ok... so this is what I gather after reading that webpage and doing some other research: A class is a collection of "things" and the universe \(U\) is the class of all those classes which share some sort of rule or rules. Is this a correct definition of a set? Let \(A \subset U\). We say that \(A\) is a set if \(\exists B \subset U\) such that \(A \in B\).
The definition of a set is that it obeys the set axioms. But if U is the class of all sets, and B is any subclass of U (and some subclasses of U will by happenstance obey the set axioms), then if A is an element of B then A is an element of U. So \(A \in B\) implies \(A \in U\) (U is usually written as V), and thus A is a set. That still doesn't define what a set is -- you need other axioms for that. Alternate set axioms for those who are short on time. Axiom of Extensionality http://us.metamath.org/mpegif/axext4.html [*] Set x equals set y if and only if for every set z, z is an element of x if and only if z is an element of y. Axiom of Replacement http://us.metamath.org/mpegif/cp.html [*] If z is a set and F(_,_) is a logical function connecting two sets, there is at least one set w such that for every element x in z, if there exists a set y that make F(x,y) true then there is a element of w which also makes F(x,_) true. Axiom of Power Sets http://us.metamath.org/mpegif/pwex.html [*] If A is a set, the class of all subclasses of A is also a set. Axiom of Union http://us.metamath.org/mpegif/uniex.html [*] If A is a set, the union of all its elements is also a set. Axiom of Regularity http://us.metamath.org/mpegif/zfregs.html [*] If A is a non-empty class, then at least one element of A has a empty intersection with A. (Thus every non-empty set contains a set disjoint from itself.) Axiom of Infinity http://us.metamath.org/mpegif/inf5.html [*] There is at least one set which is a proper subset of the union of all its elements. Axiom of Choice http://us.metamath.org/mpegif/ac7g.html [*] If R is a set (of ordered pairs), then there exists subset of R which maps (as a function) to the right side of the relation R.
Ok... so let's see if I get this now (I spent too much time on this and had to sleep on it). A class is just a "collection" of "things." A class is called a set if it adheres to all the set axioms. Right? The book hasn't mentioned that anywhere. If that's correct... I can live with it.. but, I can't really find a formal definition of a class. Maybe there isn't? I mean, we can't have a proof and definition for everything. I just find it strange that the authors of this book do not really explain what a set and class is... yet they go on to prove via the axiom of separation that not every class is a set. Huh? The proof makes sense to me, but only because I know that set and class aren't the same thing. Like saying that every apple is a fruit, but not every fruit is an apple. I don't need to know what a fruit is to know they aren't the same if I'm given an axiom about apples and fruits.
When you deal with a book called X Theory, it's all about the theory of X. Often X is treated like a primitive construct within a model. So defining what a set is outside of the model is not really in the book's chosen territory.
Since you understand Rudin's book on analysis, may I suggest http://www.amazon.com/Introduction-Independence-Studies-Foundations-Mathematics/dp/0444868399 Kunen does set up the foundations of set theory on a firm basis.
Well, yes. That makes sense. Had I asked a lot of questions about arithmetic when I was learning it, I think I would have just confused myself. Thank you for the help... I think I now have the general idea. I'll check that out. Thank you. Why? Shouldn't we then be required to define the terms used in the definition of the primitives? At some point you've got to stop. When you learn 1+1... was it necessary to learn what "1" and "+" mean? You had a basic idea without having to define them. I recommend you read Bertrand Russell's book Introduction to Mathematical Philosophy if you want to get an idea of the level of complexity involved when defining what a number is and what "+" means. Of course, things have advanced since that time. But, it's a very challenging read.
