PoW 51% attack vs. BFT 1/3 attack? So from what I understand, Bitcoin's PoW is prone to 51% attack, but as a distributed system it is also prone to BFT's 1/3 attack right? I think it's mathematically proven that in a distributed system, if you have more than 1/3 bad nodes collaborate together, then you cannot safely reach a correct consensus no matter what?

So for Bitcoin, there are two types of possible attack scenarios, one is if a miner node consistently has more than 51% computing power of the whole network then it can double-spend indefinitely and basically has unlimited cash to spend. 

The other is if more than 1/3 of the nodes in the whole network are bad guys working together with modified malicious node code then they can potentially stop the correct block being accepted and make it impossible for others to know which is the correct longest chain, thus prevent valid transactions being processed and recorded.

So are my understandings correct?
So from what I understand, Bitcoin's PoW is prone to 51% attack, but as a distributed system it is also prone to BFT's 1/3 attack right?


No. Bitcoin is not a "consensus system" by any of the traditional definitions: It never reaches a state from which it can't, at least, theoretically roll back. In theory today's blocks could get undone years from now, but in Bitcoin the computationally difficulty of that increases over time.

As a result the impossibility results for consensus don't generally apply.


  The other is if more than 1/3 of the nodes in the whole network are bad guys working together with modified malicious node code then they can potentially stop the correct block being accepted.


Nah. So long as an attacker does not manage to partition the graph of honest nodes, they could control (say) 99% of the nodes and the network would continue to run just fine. There isn't some magic threshold number of nodes where partitioning happens,  you need only have a single honest peer to be completely functional.