* - don't bother if the cabbage is the white limp variety - your slaw won't be worth the makings.
Cut the cabbage in quarters, place in a large bowl of well-salted cold water, and let stand for at least 1 hour. This is to make sure any possible "critters" emerge - nothing spoils good slaw like a many-footed friend. Once soaked, drain and place the cabbage on a large chopping board and chop away like mad. The cabbage must be very finely shredded. Dump into a big bowl and shred the apples, celery, and carrots. Add to cabbage, stir in celery seed, salt, and mayonnaise, and blend well.
Refrigerate at least 2 hours before serving to allow flavors to mellow. If slaw is to be transported to fish fry site, keep very cold - mayonnaise mixture can be deadly if allowed to stand too long at room temperature.