Different types of Set Theory

I have just now learned that there are different types of Set Theory. I read Naive Set Theory by Paul R. Halmos, but other than that I have no other knowledge of other...set theories.

Could anyone explain the difference between Naive Set Theory and Axiomatic Set Theory? I thought they were the same thing, but apparently that is not true. Or possibly provide sources to research this topic?

What about NBG and ZF set theory?

I originally thought that Naive Set Theory was simply the topic of Set Theory, but for introductory level. It relies on axioms and such. How then is this different than axiomatic set theory?

Solution 1:

Naive set theory is some general term for set theory where axioms are not thoroughly introduced and studied. We describe some properties of sets, and usually go out with the (provably inconsistent) comprehension axiom schema which essentially says:

Every definable collection is a set.

But as the parenthesis remark points out, that axiom schema is inconsistent. One can construct all sort of collections which cannot be sets. In naive set theory we casually toss this worry aside, and rely on the fact that the sets we are going to meet are going to be sets. By that virtue, naive set theory is often concerned with finite sets, countable sets or with "arbitrary sets" which are assumed to exist.

Despite its name, and its sore philosophical limitation of being inconsistent, you can develop quite a lot of mathematics within naive set theory. This is good because this development can be later carried out formally in the [not yet inconsistent] axiomatic set theory.

So what is axiomatic set theory? In axiomatic set theory the student first has to have some basic knowledge of logic, and we begin by writing down axioms. Then we prove from these axioms all sort of theorems and deduce more and more information. These axioms describe in a formal language what properties we expect sets to have. For example, we expect that if $X$ is a set then its power set is also a set itself. So we can formalize that as an axiom.

Axiomatic set theory is concerned with what statements we can prove from what axioms. For example, one cannot prove that $A\notin A$ for every set $A$, without appealing to the axiom of foundation (or some variant thereof). The proof of this unprovability is a common, and very nice, exercise in axiomatic set theory books and courses.

Where naive set theory is often given as some general outline to how sets should behave, and some basic understanding of the connection between set theory and general mathematics; axiomatic set theory investigate the sets themselves. Things like the axiom of choice, the continuum hypothesis, cardinal and ordinal arithmetics, infinitary combinatorics, and so on. These have applications to general mathematics outside of set theory, but those are still investigated for their own sake.

Finally, both naive set theory and its axiomatic version come in "flavours", but whereas naive set theory is mild, in the sense that one pays less attention to assumptions such as "real numbers are sets" or "real numbers are not sets", and so on; in axiomatic set theory one pays very close attention to the starting assumptions and the language in which one expresses these assumptions in.

For example, $\sf ZFC$ which is one of the common (if not the common) set theories is stated in the language where objects are sets, and we only have $\in$ to define things (from that we can define $\subseteq,\varnothing,\mathcal P(\bullet)$ and so on, but formally we only have $\in$ and $=$). On the other hand, one of its extension $\sf NBG$ - which you have mentioned - lives in a slightly larger language which allows objects which are not sets, called proper classes to exist in a "meaningful way" (whatever that means). There are subtle differences and similarities between these two theories. Often in axiomatic set theory we study extensions of $\sf ZFC$.

There are other theories based on a whole other approach to sets, like theories in which the basic notion is $\subseteq$ rather than $\in$; or theories where the atomic notion is "function" rather than "an element of". All these are very very different set theories, even if sometimes we can prove they end up proving "pretty much" the same statements about sets. These are also topics of axiomatic set theory, even if less mainstream and conventional.

Solution 2:

Naive set theory is enough set theory to get by in most of mathematics, but suffers from a number of unfortunate paradoxes.

In naive set theory, we call any collection of elements a "set." We don't say what the elements are, nor do we place any restrictions on which elements can come together to form a set.

However, this simple view has some problems, one of the most famous of which is Russel's Paradox:

Does the set of all sets (call it $S$) which do not contain themselves, contain itself?

If $S\in S$, then $S \not\in S$, and vice versa: a contradiction.

Axiomatic foundations of set theory (the most common of which is ZFC) resolve this and other paradoxes by providing a rigid set of axioms, and specify, for example, what the "elements" of sets are. In ZFC, in fact, elements of sets are just other sets. ZFC also poses restrictions on what kind of sets you can form, preventing things like Russel's paradox from happening.