0. Mathematical notation

We start our course in set theory. But… why do we study set theory? Why it is here where the whole higher mathematics starts?

I think that there are many answers to those questions. Set theory simply is in the heart of mathematics. It supplies the rest of maths with notions and assumptions. It gives mathematics its language. So that mathematicians are able to communicate in a precise way, which is easy to understand why such a precision is crucial in mathematics. Mathematicians achieved it fully (somehow in contrast to layers, where more precision is also needed…).

But there is more to that. Set theory is also a discipline which has elaborated perfect mathematical reasoning. Precise, logical, but also amazingly ingenious and spectacular reasoning. Ability to create such a reasoning, which one can improve studying set theory, seems to be the most basic ability of a good mathematician. And, I dare say, the most basic ability of an intelligent human.

Finally, the most elusive argument. Set theory magnificently shows the greatness of human mind. It describes an elusive part of the world, going on in a human mind. But it is not just thinking up things, it is discovering them. It feels extraordinary to be able to discover something using only your own mind.

What do we discover? We discover phenomena the most elusive matters, finding answers to the most extraordinary questions. Questions about infinity, about the sense of logical reasoning, about the nature of the world. The answers are absolutely rational and metaphysical in the same time, and show that between these two aspects there is no contradiction.

This first class is devoted to introducing mathematical language, first attempts to proving things and dealing with sets.

Main notions:

  • \forall_{n\in\mathbb{N}} means “for all natural numbers n“, \exists_{m\in\mathbb{N}} means “there exists a natural number m“.
  • so \forall_{n\in\mathbb{N}}\exists_{m\in\mathbb{N}} m=n+1 means “for all natural numbers n there exists a natural number m such that m=n+1“.
  • to prove the above statement we should find such m for any given n, so for given n we let m=n+1 and we are done.
  • to disprove a statement we usually show a counterexample, e.g. to \forall_{n\in\mathbb{N}}\exists_{m\in\mathbb{N}} m=n-1, we set n=0 and notice that there does not exist m\in\mathbb{N} such that m=n-1.
  • \{n\in \mathbb{N}\colon 2\mid n\} means the set of all natural numbers n which satisfy the given condition (in this case n is divisible by 2).
  • operations on sets:
    • A\cup B — union of sets A and B — all the elements which are in A or in B
    • A\cap B — intersection of sets A and B — all the elements which are in both A and B
    • A\setminus B — difference of sets A and B — all the elements which are in A but not in B
    • A\triangle B — symmetric difference of sets A and B — all the elements which are exactly in one of the sets A and B.
  • you should get used to the situation that the elements of sets are sets. Actually in mathematics it is always the case.
  • we can denote finite sets simply listing their elements, e.g.: \{2,3,5\}. Sets do not have anything like order or multiplicity, so e.g.: \{2,3,5\}=\{5,2,3,2\}.
  • We say that something is an element of some set denoting it by \in. E.g. 3\in\{2,3,5\}.
  • Set A is a subset of a set B, if every element of A is also in B. We denote it in the following way: A\subseteq B, e.g. \{2,3\}\subseteq \{2,3,5\}. We say sometimes that A is included in B.
  • E.g.: set \{2,3,5\} has 3 elements: 2, 3 and 5, and 8 subsets: \varnothing (the empty set), \{2\},\{3\},\{5\},\{2,3\},\{3,5\},\{2,5\},\{2,3,5\}.