Place the ham bone in a large pot over high heat. Add water to cover, bring to a boil, reduce heat to low and let simmer overnight. Allow to cool and clean the stock, discarding the bone, excess fat, and any cartilage. Return any good ham meat to the pot and bring back to a boil.
Add the onion, garlic and ground black pepper and allow to simmer for 1 hour. Then pour in the split peas and the carrots and allow to simmer for at least 1 more hour to allow the peas to break down. Stir together well and season to taste.
