If a validator signs two blocks at the same height, they will be slashed, lose their tokens, and not be able to participate in consensus. They get slashed when anyone submits evidence of them doing this. This trivially solves the "nothing at stake" problem from 2014 that you wrote your long original comment about.

If I understand it correctly, you are now saying that someone would DDOS the entire gossip network, completely halting any more production of blocks so that their slashing doesn't go through?

We're not even talking about "nothing at stake", or anything having to do with PoS anymore. We're just talking about a massive DDOS of an entire network. Node operators in PoS networks, as well as Bitcoin, have ways of dealing with DDOS which are the same as how anyone deals with it, and I don't need to get into them here.

If someone was able to overcome these DDOS mitigations and completely prevent a PoS network from receiving any legitimate transactions, they could do this to Bitcoin as well.

> They get slashed when anyone submits evidence of them doing this.

Who will record this "evidence" to the blockchain? Anyway there will be two versions of the blockchain. In one of them attacker's stake was not slashed and there is no any "evidence" of his malicious actions.

The "evidence" is two blocks, with the same block height, both signed by the attacker.

Not sure what these two versions of the blockchain you're talking about are. Signing two blocks at the same height with the same chain id is the slashable offense. It doesn't matter what's in them.

If the attacker wants to have his own blockchain off in the corner where he has all the money, nobody cares.