There's a couple of ways that are good indicators. You can look any of these up for more.
1. There is probably a wear indicator in the tire itself. A small, rounded divot right in the center of the tire, look for it. If it's invisible, your tire is worn out.
2. You start to see threads underneath any part of the rubber.
3. You flat a few times on the same tire.
4. The tire starts to "square". That is, you look at a new tire straight on, and it's round. A "squared" tire isn't actually square, but you can visibly tell it's not round anymore.
Anything else should be pretty obvious (big hole, bulge, etc.)
6,000km is a LONG way to ride on any tire, even on the best roads. Most riders swap them out about half of this. On performance tires, such as Continental 5000's for road bikes, it's more like 2,000km. On the Specialized Turbo cotton, even less. You'll simply start to flat, a lot.