What is the logical reason to use a proof by contradiction Given an arbitrary set $A$ of real numbers. I want to decide if the set $A$ is infinite. 
My question is: What is the logical reason to use a proof by contradiction.
I can think for example that there is no known method to prove this directly. But I am not convinced with this reason.  

I would say the 'logical', or at 'least conceptually 'intuitive' reason for using a proof by contradiction is because you are trying to prove that something is not the case, i.e. some kind of negation. 
Now, in this case, we can naturally think of 'infinite' as 'not finite'
Of course, there are more 'positive' formulations of infinite as well ... and so why would 'infinite' be the 'negative' one, and 'finite' be the 'positive' one? Still, I think the more natural way to think about 'infinity' is in a 'negative' way.
Moreover, we can do a lot more with something being finite. Being finite, allows us to say: ok, so it is some number ... and we're off to the races trying to see what follow from that.... hopefully a contradiction!
