I wear a belt on darn near everything and you probably will have to look hard to find someone who can out lift me over a month's worth of work that doesn't wear one.
Training without a belt isn't going to give you stronger abs or "core"...training your core will. Adding a belt isn't going to automatically add 30 pounds to your lift.
If you want to be strong...pick up heavy freaking weights with good technique, move it properly, wear whatever support gear necessary, and train your abs when you are done.
Over the last 2 years I've added about 200 pounds or more to my squat, more than 100 pounds to my deadlift, and a good 40-50 pounds on my bench press. What does wearing a belt, or other support gear DO for me? It allows me to move heavier weight week in and week out while others are having to "take a break" so that their lower back and "core" can recover. When I need stronger abs...I do my sit-ups, full ROM, with a 45 pound plate behind my head for several sets and reps...