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Dec 22, 2015 at 2:25 comment added abeikverdi Yea its important to distinguish double spends and invalid transactions and I think the question didnt make that clear. "Even if only one miner is honest the valid chain will continue." but the valid chain is not accepted by the network since it doesn't have the valid proof of work. Its like you maintain a valid blockchain on your own but no cares about you! The valid chain is not valid until it gets the proof of work done. It is also gonna be an issue for SPVs since they only follow the longest chain that has put work in it.
Dec 21, 2015 at 19:21 comment added Jannes "assuming more than 50% of the miners are honest that false block will be orphaned." They don't need to assume more than 50% is honest. Even if only one miner is honest the valid chain will continue. The invalid blocks are just completely ignored, so that chain won't grow at all. This is when talking about invalid blocks (wrong signatures, block reward too high, etc.). Note that double-spends are NOT invalid transactions. Both transactions ARE valid, but only one of them ends up in whatever will be the final blockchain.
Dec 21, 2015 at 8:34 history answered abeikverdi CC BY-SA 3.0