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When we talk about countable sets, we're discussing a concept in mathematics that pertains to the size of a set. A set is considered countable if its elements can be matched one-to-one with the set of natural numbers. In simpler terms, you can 'count' the elements, even if there are infinitely many of them, by aligning each one with a natural number.

Two classic examples of countable sets are the set of all natural numbers itself, and the set of all even numbers. At first glance, one might think that there are more natural numbers than even numbers since the even numbers are a subset of the natural numbers. However, because we can match each even number with a natural number (e.g., 2 with 1, 4 with 2, 6 with 3, and so forth), both sets are actually the same size in terms of their 'countability'. This fascinating property underlies some astounding results in set theory, like the fact that two countable sets can be combined to form another countable set.

The heart of understanding countable sets lies in the idea of one-to-one correspondence. This is a way to pair each element of one set with exactly one element of a second set, such that there's no overlap and no element is left out of either set. Imagine two rows of boxes where each box in one row is connected by a line to a box in the other row — that's one-to-one correspondence in a nutshell.

For instance, let's say you have a set of chairs and a set of people. If you can seat every person in a chair without any empty chairs and without any person standing, you've achieved a one-to-one correspondence between people and chairs. This concept extends to infinite sets as well, and it's what allows mathematicians to determine whether an infinite set is countable or not. It's the technique used in establishing a one-to-one correspondence between the cartesian product of two countable sets and the natural numbers, which helps in proving the countability of the combined set.

The natural numbers are the simplest numbers we use: 1, 2, 3, etc. They're what we use to count objects and are fundamental to arithmetic. In set theory, they hold a special place because they form a 'simplest' infinity, known as 'countable infinity'.

Every natural number has a successor (just add 1!), and this goes on endlessly. This property gives the natural numbers their countability, as they can be put into a sequence where each number is followed by its unique successor. They provide an essential baseline against which the 'size' of infinite countable sets is measured. If a set can be put in one-to-one correspondence with the natural numbers, it's said to be countably infinite. This ties back to the exercise problem where the cartesian product of two countable sets is mapped out to show it's countably infinite as well.