Whether you're eating it raw or cooked, your classic head of cabbage (Brassica oleracea) is not a particulary good source of antioxidants. Part of that is our own fault, because most of the nutrients are in the dark outer leaves which of course, we almost always discard during preparation. Though even if you kept those intact, this is one of the least nutritious cruciferous vegetables, broccoli and kale score much higher.
This however doesn't mean you should give up on cabbage. It is good healthy choice just so long as you choose the right variety. Even after its boiled, red or purple cabbage has ORAC values which are 300-400% of regular cabbage. This is due to the anthocyanins - a class of antioxidants which is found in their vivid purple leaves.
Even if you stick with the plain Jane variety, it's such a low calorie veggie that it's a tasty filler to use in recipes, in order to reduce total caloric content of the dish. You could eat an entire head of it (medium size) and you would only be consuming 223 calories and perhaps to your surprise, 12 grams of protein. Given that the typical serving size may be 1/4 or 1/8 of a head, it's a guilt-free and cheap ingredient to include.
USDA Database for the Oxygen Radical Absorbance Capacity (ORAC) of Selected Foods, Release 2 - Prepared by Nutrient Data Laboratory, Beltsville Human Nutrition Research Center (BHNRC), Agricultural Research Service (ARS), U.S. Department of Agriculture (USDA) - May 2010