He could be tail biting, especially if moving his tank to a new location was stressful for him. Or it's possible that during the move, he got caught on his plastic plant (I'm assuming that's what you have in there) or decorations and tore his fins. But overall, torn fins are not usually a cause for concern unless it's fin rot which this doesn't seem to be. Warm, clean water is the best way for fins to heal.
Also, water that is too cold can cause stress. Bettas are tropical and need warm water, ideally between 76-82 (depending on who you talk to, but generally, they all agree that anything below that is too cold for a Betta). A half gallon will work for a while if you are doing at least 3 - 4 100% water changes a week (I speak from experience here), otherwise, you are risking the health of your fish. The general rule for all fish is 1 gallon minimum per inch of fish, so if your fish is 2 and 1/2 inches long, you will need at the smallest a 2.5 gallon tank, sometime in the nearish future (but you can keep him safely in the 1/2 gallon tank if you do multiple cleanings a week for a while until you can get a bigger one, but you do need a heater)
Last edited by sainthogan; 11-25-2012 at 12:53 AM.