# Sergei Yakovenko's blog: on Math and Teaching

## Sunday, October 27, 2013

### Lectures 1 and 2 (Oct. 22 & 29, 2013)

We start our course by first carefully walking in a zoo with several surprising, wonderful or dangerous beasts, all caught in the Jungle of Infinity. To tame them, we need to wear some protective gear made of ironclad formulas, operate from the safety of well defined sets and use the tools provided by functions 😉

The lecture notes (considerably updated and revised in comparison with the previous years) can be found here. Please report typos, errors and complain about obscure instances in the comments.

Please pay attention to the problems scattered over the text: they are mostly for self-control, but if you are not sure, you may write your solutions and submit them (e.g., in the comments to this post, but also directly to Inna, if you want to keep it private).

## Infinity: first accurate steps

1. Finite and infinite subsets of $\mathbb N$.
2. Admissible infinite operations: infinite unions and intersections. Quantifiers.
3. “Small” and “large” infinite subsets of $\mathbb N$.
4. One-to-one maps as the means of comparison between various infinite sets.
5. The first “paradox”: $\mathbb N\times\mathbb N\simeq\mathbb N$.

## First Encounter with Infinity

In the first lecture we discuss the dangers that are inherently present when we transcend our finite intuition and consider infinite quantities, constructions etc.

The first step to do is to clean up our language, restricting it to the most transparent and unambiguous grammar and vocabulary. This is the language of the sets and operations on them, and logical formulas involving quantifiers, to formulate meaningful (true or false) constructions.