Ds has asthma, and when he was about 2, it got worse, with twice daily steroid inhalers and 4-hourly others. I vacuumed at least twice a week, with a Roomba every day. (Roomba useless for pet hair, btw - only have it as dh loved the excuse, and fine for surface dust.)
Gp was happy to give stronger inhalers, but no tests, so we went private. I really wished we had done that before. It confirmed that he had no cat / dog / dustmite allergies, so "just" asthma. We were advised to try a new treatment, which is one tablet every morning, and could stop inhalers except for when he has a cold, when it helps with general breathing. In retrospect, I should have done this earlier.
So, whilst I think you cannot keep your cats if your child is really, seriously allergic, get your son tested as then you'll know.
PS: Someone said earlier in thead (Mother... something?) that some exposure to allergens is beneficial long term - I think she is right. I don't have documentation to hand, but our consultant, who is a world renowned specialist on allergies, said that, as long as it is not a serious allergy but rather sensitivity, limited exposure is good, as your body builds up resistance. Uhm, or something, I can't exactly remember the words, but you get the idea. Real life example of sensitivity is me - when we got our kittens, my skin would itch incredibly, even though I washed the kittens every two weeks, was really clean, followed advise on managing. It took about 6 months for it to stop happening. Similarly, ds became lactose intolerant when he was smaller. Paediatrician recommended lactose free for 3 weeks, then very, very slowly adding lactose again until full strength 3 months later. Again, if serious allergy I don't think you can risk it, but where intolerant / sensitive, cutting out completely leads to permanent situation where it might have been temporarily otherwise. There were some nut trials (obviously life threatening, so not even remotely suggest doing this yourself!!) done last year, weren't there? I think results there was that some participants were able to have a peanut etc for the first time in their lives. It would make life, where there could potentially be nut contamination, considerably less stressful, I would imagine. Sorry, got a bit sidetracked there...