Today Jenny and I watched Bender's Big Score. I had to watch it a couple times to understand a couple things, and I've come to realize a problem in the story line. Don't read on if you don't want spoilers.
Fry goes back in time to the year 2000 because he is going to be killed by the scammers. This is a duplicate Fry (call him Fry A, which later becomes Lars). According to the non-paradox equation, duplicates are doomed. That is why Fry A leaves Leela at the altar and kills himself to save her later. Fry A then attempts to re-integrate into the year 2000, finds he is hungry with no money, and decides to go back to the Cryogenics lab to get the pizza. When he discovers the pizza is cold, he decides to go back in time an hour to get the warm pizza. This duplicate is Fry B, who eventually falls into the cryogenic chamber with the original Fry. Because he is a duplicate, he is doomed too.
So, what happened to the original Fry? He went back in time and became Fry A. Unfortunately, that means he ceased to exist in the year 3007. That means, after Fry goes into the time sphere in 3007, any Fry's that exist are duplicates. We already know that Fry A knows he is a duplicate, is doomed, and dies.
But what about Fry B? After he is unfrozen in 3000, he re-enters the cryogenic tube for 8 years, and he shows up during his own funeral. Fry B is a duplicate, so by definition he is doomed. However, as the story plays out, this is the one Fry that survives. This is a paradox.
Consider a simpler example. A scammer goes back one day and meets himself. The original (A, who was the scammer being met AND the scammer entering the time sphere) and the duplicate (B) return just after A uses the time sphere. B is doomed, and dies when the smell-o-scope falls on him. But what about A? If he was with B until just after the use of the time sphere, then A would never have used the time sphere. This is a paradox.
However, this is a different paradox that Fry's case. Neither Fry A or Fry B prevent the original Fry from using the time sphere, so there is no impossible case.
So what should have happened? And all original time travelers should cease to exist from the first time they use the time sphere and onward. Any travelers that exist chronologically after the use of the time sphere are duplicates, which are doomed. Hence the scammer time travel series of events should never have happened. It wouldn't even work if they showed up together and scammer A entered the time sphere, otherwise there would be an infinite number of scammer Bs. Finally, Fry B, who is the only surviving Fry in the end, should be doomed.