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What does Zermelo–Fraenkel set theory mean?

According to Wikipedia, Zermelo–Fraenkel set theory is a set theory that is proposed to overcome issues in naive set theory. I really appreciate if somebody can kindly explain the basic concepts involved in this Zermelo–Fraenkel set theory so that high school students can understand.

In mathematics we have an idea, some concept. For example the idea of "set" was modeled after the idea that two collections are equal if they have the same members.

If I list your family it doesn't matter if I'm listing your father before you mother, or vice versa. I am only interested in the members. Similarly if I listed exactly the members of your family then this is a list of your family.

So sets were models after this idea. But we want more from sets, we want them to have certain properties. And at first we assumed these properties must always hold, but it turned out that there is an inherent contradiction in naive set theory.

So mathematicians of the early $20^{text{th}}$ century suggested axioms, which are strict rules that the mathematical objects called "sets" must obey. Amongst those mathematicians there were Ernst Zermelo and Abraham Fraenkel, after which the theory is named.

Before we start, let me just add one definition. Given two sets, $A$ and $B$ we say that $A$ is a subset of $B$ if all the members of $A$ are members of $B$. For example, all the students in your class are also students in your high school, so the set of your classmates if a subset of the set of your schoolmates.

The axioms of the Zermelo-Fraenkel set theory describe the properties we expect sets to have, in a mathematical way. The incredible part is that our language only contains one binary relation symbol, $in$, which is the membership symbol. Now remember that set theory only talks about sets, so all our objects are sets, including their members. This means that the most primitive notion is the question "Is this set a member of that set?.

This may sound baffling but we can encode numbers by sets and define all the mathematics within this universe, all this with only the membership relation to work with. The axioms tell us that sets have the following properties:

1. Two sets are equal if and only if they have the same members.


2. If we have a set, then the collection of all of its subsets is another set.


3. If we have a set of sets $I$, then we have a set $U$ which is the union of those sets, namely all the members of $I$ are subsets of $U$, and $U$ is the smallest possible set with this property.


4. There exists a set which has no members.


5. If we have a set $x$ and a property $varphi$, then the collection of all the members of $x$ with the property $varphi$ is a set.


6. If we can describe a function whose domain is a set, then its image is a set.


7. There exists an infinite set$^*$.


8. In every set $x$ which is not empty, there is a member $y$ which does not have any members shared with $x$.

These may sound a bit strange, but they serve a function. They tell us that some sets exist (empty set, infinite sets); they tell us how to create new sets from old sets; they tell us when two sets are equal; and they give us some strange condition on the membership relation (the last axiom).

It should be pointed that axioms 5 and 6 are not really axioms. They are schemata of axioms. This means that those are actually infinite lists of axioms which are easily described formally, and a computer can easily verify whether or not a certain sentence is an axiom from these lists or not.

Amongst the contradictions which these axioms solve is the one given by Russell, which is described by the collection of "all sets which are not members of themselves", this collection cannot be a set -- although we can describe this collection. With our axioms we have that if this collection is a set then we have a contradiction, so this is not a set to begin with.

But the axioms are important, because they give us a rigid framework for sets. Otherwise we just think of sets naively and so every collection we can describe must be a set, but Russell's collection cannot be a set. And that is the contradiction.

$^*$ I slightly dumb down this axiom, we require slightly more than just an infinite set, but it is easier to understand it this way. What we do require is that there is a set which "looks" like the collection of the natural numbers.

By 'poking around' on wikipedia and elsewhere (like here, math.stackexchange), a person with mathematical talent will be able to get a 'layman's appreciation' of how Zermelo–Fraenkel (formal) set theory overcomes issues in naive set theory.

While perusing this topic (also as a 'layman'), I found the following passage from wikipedia,

$quad$Article: Axiom of regularity
$quad$Section: Regularity, the cumulative hierarchy, and types

of interest, especially since it's all in English.

Dana Scott (1974) went further and claimed that:

The truth is that there is only one satisfactory way of avoiding the
paradoxes: namely, the use of some form of the theory of types. That
was at the basis of both Russell's and Zermelo's intuitions. Indeed
the best way to regard Zermelo's theory is as a simplification and
extension of Russell's. (We mean Russell's simple theory of types, of
course.) The simplification was to make the types cumulative. Thus
mixing of types is easier and annoying repetitions are avoided. Once
the later types are allowed to accumulate the earlier ones, we can
then easily imagine extending the types into the transfinite—just how
far we want to go must necessarily be left open. Now Russell made his
types explicit in his notation and Zermelo left them implicit.
[emphasis in original]