This is a post on one aspect of the infinitely muddled and confusing (confused?) concept of cardinality.
Take the product and box topologies of . The product topology is first countable, whilst the box topology is not. A good discussion is given in Munkres. I will discuss this with an analogy.
Let us take a man who lives forever. Also, let us take an arbitrary day in his life, and call it Day 1. Everyday after day 1, he throws one ball into an infinitely large basket. Will the basket ever have balls? Remember that the man will live forever, and will keep on throwing balls into the basket long long long after we’re dead.
Intuition says “yes”. For any given number , we can show that the basket will contain more than balls after days. However, showing that the basket will eventually contain more balls than any finite number does not show that it will eventually contain an infinite number of balls. This is very very important. One might think . Surely this is common sense. However, the point to note is that even if the number of balls will be greater than , it will still be finite. Say we take and check the basket after a trillion days, although the number of baskets will be greater than , it is still finite (a trillion).
It is a valid mathematical argument to show that something is infinite by proving that it is greater than any arbitrarily chosen finite number. However, here we are choosing first , and then determining the set which contains more than elements. This is an invalid procedure. Looking at it from another perspective, whatever set we choose, we can show it to contain only a finite number of elements. Hence, the basket will never contain balls.
My contention with this is: what if we can go to ? We will surely find balls in the basket then!
We can’t. Not because we can’t do this in the practical world. But because we just can’t do this according to the current axioms in Mathematics. If there were such an axiom (let us call it Axiom A) which would allow us to do so, then we would surely find balls in the basket. But then we’d have to say “we can find balls assuming Axiom A.”
PS: I really hope Axiom A catches on. The results obtained from Axiom A are so much more intuitive!!