First of all- the definition is not mine. It is what is in the books. The terms "recognize" and "recognizable" is defined in the context of complexity theory. So are the terms "decide" and "decidable". "Accept" is not. So when you say you don't think some definition is good please read the exact definition first. There are many serious and great mathematicians and computer scientist involved in that theory and it is irrespectable to them to comment it without being familiar with it.
Second - What I meant to say in my first comment is that the way P?=NP problem is defined it is provable. The fact that it is provable can be proved easily and I will leave this as exercise to you.