In one of my previous posts, I wrote a very simplified explanation of the Banach-Tarski paradox, a paradox that creates two identical spheres by partitioning a sphere into a set with a finite number of elements and a finite number of rotational motions. (link:https://www.seanyoonbio.com/post/math-evenings-the-banach-tarski-paradox)
In the post, I haven't got into much detail about the mathematics, partly because the article word limit is impossible to encapsulate the mathematics behind it, and partly because I myself was incapable of totally understanding the mathematics. But now that I am more knowledgeable about group theory, I think now I can display a rigorous proof of the theorem. Since the proof is very long, I will post in 5-6 separate posts.
But before we move into the proof, it is essential that we set the definitions of some key terms. Terms like group--we've previously defined them. (link: ) But what is infinity?
One important notion to note is countable infinity and uncountable infinity. Aren't all infinity the same? Think about this. There are infinite natural numbers, making the cardinality of the set N infinity. In the case of integers, there are negative numbers as well, and 0, so there are infinity*2+1=infinity elements. But what about real numbers? Between every integer, there are infinitely many real numbers. Let's zoom in the number line, and try to get the closest two real numbers between 0 and 1. But no matter how much we zoom in, it will be impossible to find the closest two real numbers. Hence, this must be a different "type" of infinity, and this is what we define as uncountable infinity.
We will define these more rigorously.
Def. A countably infinite set is a set such that
the set is not finite and
there exists a bijection between all elements of the set and the set of natural numbers.
Exp. A more intuitive way to think about this is "are the elements able to be lined up in some rule?" For instance, natural numbers can be lined up as "1, 2, 3, ..." In the case of integers, "0, 1, -1, 2, -2, 3, -3, ..." Finally, in the case of rational numbers between 0 and 1, "1/2, 1/3, 2/3, 1/4, 2/4, 3/4, ..." If these elements are able to be "lined up," we can manufacture a bijection from this sequence to N, and hence the set must be countable infinity.
In contrast, any infinite set that is not countably infinite would be an uncountably infinite set.