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This lecture was dedicated to the first four axioms of Zermelo-Frankel set theory.
But what is ZF set theory in the first place?
All of us have an intuitive idea of what does the word "set" mean.
We often think of it as of collection of selected objects to match some property.
One can think of the set of all chairs in that classroom or even set of all chairs in all the classrooms of that school.
However, our language is way too powerful tool to express declarations of sets.

We can say that we want to have "a set of all sets that are not an elements of themselves", in other words,

But then we won't have an answer to the question "whether a is an element of itself"?
It brings us to the paradoxal situation:


That paradox is named after Bertrand Russel ( http://nn.lv/de ).
Many paradoxes and contradictions have forced mathematicians to invent set theory which would be as powerful as the classical (naive) set theory but not as paradoxal.
There are two main approaches to solve that problem —

1) Defining an element of the set as a completely different object from the set so that a set couldn't be an element of any other set.
2) Strictly disallowing any set definitions which are not based on any existant sets.
The second approach appeared to be more useful and powerful.
Nowadays the set theory that is widely used is ZF set theory. The main axioms of that are described in the following article.

The first axiom is very basic axiom. It says "if two sets contain the same elements then those are equal"

It shouldn't be confused with the signature equivalence.
If we have multiple instances of the set in our formulas then
we can always say that without proving that every element of is also contained in .

Rather important that due to there may be only one empty set.

The second axiom states that there is an empty set.

That axiom is optional and can be further deduced from less powerful.
However I prefer to introduce as it can be written as a proper formula
in First Order Logic signature.
Also, having the empty set we may deduce theorems having only four ZF axioms.
If we've used the alternative axiom we'd waited for the sixth axiom.

Third one. The axiom of regularity.
In each non-empty set there is at least one element which doesn't contain any element of the set.

The last axiom which was presented on the lecture was the axiom of unordered pair.
It states that if we have two sets — and then there is a set which contains and and nothing more.

We can easily see that if we let then we'll get singleton.
Meanwhile now we can use signature at least for singletons and two-element sets.

Two theorems were proven in that lecture.

Lemma:
There is no set which is an element of itself.

Proof:

Say, is an element of itself
.
By there is a singleton b such that
.
But by there must be an element which doesn't contain an element of .

Thus, the fact that is the only element of contradicts with .

Thorem:
There is an infinite number of sets.

Proof:

By consequent applying to the we'll get an infinite number of sets:

A home task for those who hadn't attended to the lecture is to finish the proof.
To finish the proof you have to prove that .
That's all by now. Post any questions here or to sweater.very.lv.