Because if TDT endorsed the action, then other people would be able to deduce that TDT endorsed the action, and that (whether or not it had happened in any particular case) their lives would be in danger in any hospital run by a timeless decision theorist, and then we'd be in much the same boat. Therefore TDT calculates that the correct thing for TDT to output in order to maximize utility is "Don't kill the traveler," and thus the doctor doesn't kill the traveler.

TDT could deduce that people would deduce that TDT would not endorse the action, and therefore TDT is safe to endorse the action. It seems like the gist of this is: (a) i've decided that killing the traveler is wrong (based on something other than TDT) and (b) TDT should do the right thing.

I upvoted and like this post. Some of it just strikes me as magical