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.  

The reason most people do proof by contradiction isn't really logical.
When you do a direct proof, you have $n$ assumptions and $1$ predetermined statement to prove.
When you do proof by contradiction, you have $n+1$ assumptions and you succeed when you prove any false or contradictory statement.
Some people find the second scenario more freeing.  If you are interested in the subject you are proving about, it is worthwhile to unwind your proof by contradiction to see what a direct proof would look like.
