I know that my question has been answered on StackExchange for the countably infinite case. I understand why this holds. However, I'm reading "A course in homological algebra" by Hilton,Stammbach and they assert in an exercise that $V\cong V\oplus V$ holds for countable dimensional (so possibly finite) vector spaces. I can't see why or even if this claim is true.
Please help if you have any idea if this is true or not.