Why isn't this finite set $A = \vert\{x | \space 2<x<3, x \in \mathbb{R} \}\vert$

Question

$$A = \vert\{x | \space 2<x<3, x \in \mathbb{R} \}\vert $$

Why isn't this finite set?

Answer 1

It is infinite because in contains the numbers $2+\frac12$, $2+\frac13$, $2+\frac14$, …

Answer 2

Because we are working within an interval of the real numbers, which is an uncountable set. Try reading about Cantor's diagonal argument to see why.

Answer 3

I think you are being 'mislead' by the fact the 'distance' between 2 and 3 is finite. That is, you have a finite interval here ... how come there are infinitely many things on that finite interval? Well, one way to see this is to pick any number between 2 and 3, and once you have done that, pick a number between 2 and that number and then between 2 and that number, etc. etc. I think you can see that you can keep doing this indefinitely, so that gives you infinitely many points right there.