Tie the celery seeds, mustard seeds, allspice, and peppercorns in a spice bag or a piece of fine-meshed cheesecloth.
In an 8-qt stainless steel pan, combine the tomatoes, onions, peppers, and the spice bag. Over low heat, heat and stir the mixture until the juice is released from the tomatoes. Increase the heat to medium and bring to a boil. Cook the mixture, stirring frequently, until the vegetables are soft, about 20 minutes. remove the pan from the heat and remove the spice bag.
Place a sieve over a heatproof bowl or pan. Ladle the vegetables into the sieve and drain off the liquid. Discard the liquid or save it for soup stock. Press the pulp through a food mill or fine-meshed sieve to remove the seeds.
Return the pulp to the pan and stir in the vinegar, sugar, salt, and paprika. Over medium heat, bring the mixture to a simmer, stirring frequently to prevent from sticking. Reduce the heat and simmer for about 15 to 20 minutes, or until the ketchup reaches the desired consistency, stirring frequently.
To test the consistency of the ketchup, place a small spoonful on a plate. If the ketchup mounds up and does not spread, it is done. If a ring of liquid forms around the edge, the ketchup needs to be cooked for a few minutes longer.
Ladle the ketchup into hot jars, leaving ½-inch headspace. Using a plastic knife, remove any trapped air bubbles. Wipe the jar rims and threads with a clean, damp cloth. Cover with hot lids and apply screw rings. Process half-pint jars in a 212°F (100°C) water bath for 15 minutes, pint jars for 20 minutes.