Other mistake: Captain Eureka losing his finger makes absolutely no sense. He doesn't cry out in pain, there's no blood, and where the finger ended up is completely illogical. The finger didn't just sever on its own; it's shown that severing only occurs when passing through objects. Captain Eureka goes through a wall that is a good fifteen feet away from where the finger is discovered, and it is implausible to think it could have rolled there on its own.
Plot hole: When talking about rescuing the kids, Carter mentions them in the plural. But so far, all they have to go on was Zoe's birthday, so they wouldn't have known that anybody else was there, and Thorne also had not said anything about Zane.
Continuity mistake: When Jack and Lexi are in the jeep talking a car can be seen in the mirror, but when they pull over no car passes them.
Continuity mistake: When Captain Eureka puts out the fire at the pumping station, Jack's Jeep somehow gets on fire, but we are able to see where the fire went, and it didn't come anywhere near the road or the Jeep.
Factual error: Zane used ULF (Ultra Low Frequency) to communicate with Henry and notes he can "only send numbers." This is a reference to extremely low bandwidth of maximum 300 Hz transmission. However, later Henry magically creates an upgrade that allows voice communication - not possible with ULF. Also, no way Sheriff's old radio would pick up ULF band communications in the first place.