Bettabubble, is it possible to get a pic of his fins? There could be many reasons why fins develop tears. One is if your betta is a halfmoon or other type with excessively large fins. These tend to be brittle and tear easily. Another reason could be fin biting. Ammonia burns and fin rot are only two possibilities.
With ammonia burns, you will see signs of burns on the rest of the fish as well. Their breathing may become heavier and they may develop red sores on their body.
With fin rot, the edges of the fins are blackened and they will fall off in large chunks at a time. You will sometimes see pieces of fin in the tank.
I strongly suggest you purchase an adjustable 25-50w heater. The longer your guy stays at cold temperatures, the more his immune system will be suppressed and he'll be unable to fight off disease.