Put all the ingredients in a pot. Bring slowly to a boil, skim any foam that rises to the surface, then simmer until the peas are tender, about 1-1/2 hours. Remove the ham bone, cut off the meat, and dice small. Put the meat back in the soup, adding the carrots at the same time, and continue cooking until they are tender, about 20 minutes. Season the soup with salt and pepper, and serve.