Coat slow cooker bowl with nonstick cooking spray.
Add tomatoes, onion, carrots, garlic, 1/2 teaspoon of the salt, 1/4 teaspoon of the black pepper, oregano and pepper flakes. Stir to combine. Add ribs and cover with sauce. Scatter zucchini over top.
Cover and cook on HIGH for 6 hours or LOW for 8 hours.
Remove ribs and shred meat. Discard bones and stir shredded meat back into sauce; stir in the remaining 1/2 teaspoon salt and 1/4 teaspoon black pepper.
Stir half the sauce (about 4 cups) into pasta, adding some of the cooking liquid, if desired. Stir in cheese and serve. Freeze extra sauce for future.