6.1.1. Introduction

In the last lecture we saw various examples of countably finite sets. So we will continue the discussion on countably infinite sets and the plan for this lecture is as follows.

In this chunk, the professor summarizes what was covered in the previous lecture, emphasizing a transition from studying countable finite sets to countably infinite sets. Countably infinite sets are those that can be listed in a sequence, meaning their elements can be matched with the set of positive integers. This lecture is a continuation of exploring this topic by presenting examples of uncountable sets.

Think of countably infinite sets like an endless lineup of people in a queue. Just as every person in the queue can be numbered (1st, 2nd, 3rd, etc.), countably infinite sets can be matched with integers, illustrating how they can be 'counted' or indexed.

It might look like that for every infinite set, somehow we can show that its cardinality is same as set of positive integers. But the interesting part here is that is not the case and the focus of this lecture is to argue about the existence of infinite sets whose cardinality is different from that of set of positive integers.

This chunk introduces the central theme of the current lecture: the distinction between countably infinite sets and uncountable sets. While it may seem that all infinite sets can be associated with natural numbers, the lecture highlights that this is not universally true. The purpose here is to demonstrate that some infinite sets cannot be listed in a sequence as positive integers, marking them as uncountable.

Imagine attempting to count every star in the night sky. Initially, it seems feasible to number them one by one, but as you explore further, you realize there are far more stars than you can count—hinting that some collections (like certain sets) surpass our ability to list them exhaustively.

So we begin with our first set namely set of all binary strings but of infinite length. And this set is denoted by this notation {0, 1}. So; some examples of binary strings of infinite length...

The professor begins exploring a specific example of an uncountable set, which is the set of all infinite binary strings. These strings consist of sequences of 0s and 1s that continue indefinitely. The discussion signifies that while we can comprehend finite-length strings, the properties of infinite-length strings are fundamentally different, especially concerning their ability to be counted or indexed.

Consider an infinite library where each book contains a sequence of 0s and 1s. While you can easily count and organize finite books, the infinite ones have an ending—every time you try to index them, new strings emerge, showing you can’t finish counting them.

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