I was one of those people posting in the Bungie thread linked earlier. During the Beta, I had the same issues people are reporting now - game crashes, and can hard lock the PS4 Pro, sometimes even leading to a checkerboard black and white image appearing. After reading up on the issue, I checked out the Rise of the Tomb Raider demo and had the exact same experience as I was getting the Destiny 2 beta.
Which to me, makes it sound like a hardware issue. Not wanting to be royally disappointed/pissed off on September 6th, I sent in my unit for repairs (and since it was still under warranty). Got the unit back a couple of weeks later and the RotTR demo ran flawlessly. I still wasn't sure that D2 would run without issues - but I felt better about my chances.
And thankfully, for me at least, I've not had any crashes. I did have on instance where the screen went black for a few seconds while on Patrol, but it came right back and I've never seen that issue since.
If you are having this issue with a PS4 Pro, and Destiny 2 keeps crashing - try the Rise of the Tomb Raider demo and see if you are having the issue there as well. If so, contact Sony for a repair job --- it's got to be hardware related. Hopefully, with more exposure on this issue people won't have to pay for a fix if the unit is out of warranty.
ETA:
Might be hardware. Here's a DIY solution that might help, fairly time consuming though:
1. Back up your system.
2. Reset (Initialize) your system.
3. Update system post-initialization.
4. Install problematic game.
5. Update problematic game.
6. Run problematic game to see if problem persists.
7. If problem doesn't persist, restore from back up and try again.
7.5. If problem persists post-restore, delete game install and try re-installing.
8. If problem persists, contact Sony for repair.
Forgot to mention I tried this several times as well. It never worked. Honestly, I'd say save yourself the time and try the RotTR demo first. If it's happening there, I'd practically guarantee that the issue is hardware related and won't be fixed by going through those steps above. (Not saying not to try them - but that was my experience. And man that process does take a LOOOONG time)