In a small bowl mix together the olive oil, 1 tsp salt, and 1 tsp pepper.
Heat a large saute pan or dutch oven over medium-high heat.
Rub the pork shoulder all over with the oil mixture then place in the hot pan. Cook until well browned all over, about 5 minutes per side.
Transfer the pork to a 6-quart slow cooker keeping any fat in the pan.
Keep the pan over medium-high heat and add the onion, cabbage, garlic, 1/2 tsp salt, and 1/2 tsp pepper. Cook, stirring often, for about 5 minutes.
Add the vinegar to the pan and use a wooden spoon to scrape the bottom of the pan to release any stuck on bits. Cook, stirring often, until most of the liquid has evaporated, about 5 more minutes.
Pour all the contents of the pan into the slow cooker with the pork. Put the lid on the slow cooker and cook on LOW for about 6 hours or until the pork is tender.
Taste and season with additional salt & pepper if necessary then carefully transfer the pork and cabbage to a serving dish.
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