By inconsistency, I mean illogical choices and behaviors written into characters to fit the story. In this case, there are so many levels of contrivance that the whole episode is ruined.
Foremost among these contrivances is Carter's lack of reproach when Lexi gives permission for Zoe's yoga excursion. A simple, flat "You are not her parent" response from Carter is all that was needed to finish this plot thread in its tracks. However, to stretch the thread throughout the entire episode the most logical and probable response anyone would give is ignored.
The next level of contrivance is the science, or more specifically the logical extension of science. Of late, the series has been treating science as magic, presenting scientific ideas that have no foundation in reality under the rationalization that "Eureka is SO advanced, they can make it happen". For example, last episode the rationale was to present a mummy for tourism to make money; however they used anti-gravity technology to move an sealed tomb, intact, half-way across the globe. The money from tourism would be paltry compared to what could be earned from the introduction of anti-gravity into the commercial environment. Again the writer's fail to see past the contrivances needed to make this episode happen. In this case its the conveniences that prevented the severed finger from being identified, namely that the "DNA was scrambled". If this was true, the there is no chance to save either Carter or Chuck because even if GE finds a way of keeping them corporeal, their cells couldn't reproduce properly with ruined DNA replicating. The contrivance exists solely to keep the mystery going, and only makes the resolution illogical.
The next level of contrivance is the sudden interest in Lexi by all the males in Eureka. This only exists because the writers hoped to obfuscate the true identity of Capt. Eureka from the viewer. Lexi has been in the town for a while, but only now do men start showing interest -- interest as convoluted as the entire superhero plot to win her affections?
Which then leads to the contrivance of Lexi's unexplicable incapacity to see that Capt. Eureka obviously is endangering people. He blatantly exacerbated the already ludicrous "gateway security" situation, expressly caused damage to Carter's vehicle, and unthinkingly brought risk to civilians by removing safety signs. How could anyone, even if they were smitten by Capt. Eureka, not be able to see that? But Carter has to repeatedly badger Lexi until she sees the light? The woman is not stupid, but the writing sure makes her seem that way.
Next is the contrivance of hitting a button setting off the fail save lockdown. Really, my mind boggles at how ludicrous it was for "the third highest IQ" to instigate the trouble. Not to mention the ever present pout and poor portrayal made the character into a characterization. But without doing so, the story couldn't work.
I can go on: The cute use of Zoe's birthday as the clue instead of a numerical patern like a binary SOS or the nationally recognized 911, yet there was no way to know if Carter would even see the signal.
The entire sequence running down several flights of stairs from a few rats. Most people would just back up out of their way, and the rats certainly wouldn't follow them like a tidal wave. But then at the bottom of the stairs, when they all finally agree to leave, they look down either side of the passage away from the staircase?? Come again? Have they forgotten the cardiopulmonary workout they just had? Giving the writers the benefit of the doubt, they would have looked at the stairs and said "do you remember how many flights we came down?"
The "one chance cryptex" was little more than a combination lock with an astronomically high amount of possible cominations. Further, all he did was move the tumblers into position and lock unlocked itself -- he didn't push a button or pull a lever. That means there is no method for the device to know when to apply the check the settings of its tumblers for the correct sequence. Follow that to its logical conclusion, there is no situation that the tumblers are checked to be in the wrong configuration, so there is no foundation for Thorn's fear of the lock fusing. And how convenient it was for Zane to have thought to bring climbing gear when he had no idea he'd be encountering an elevator shaft. And then Thorn leaves Zane, with not supervision, to climb down on his own and instead heads back to town? With how zealous she's been to discover the secret?
How about the fact immediately after using the phasing suit only once Carter begins phasing uncontrollably but it took longer, several times in fact, for Chuck? Or why didn't the suit lose coherency as well? Or why when cornered at the hospital did Capt. Eureka decide to run through Jo and Carter instead of going the opposite direction, the direction he phased in on?
Or the fact that not only is there a giant biodome underneath Eureka, but a vast military complex that has been forgotten about as well? That's avoided notice by a burrowing vehicle that's been criss-cossing underneath the town. And the climate/sewage network that lurks under every building in Eureka -- including the local high school that was built right next to it. No one, not an engineer or construction worker or anyone noticed underground pipelines, wires, or even the metal wall already there?
I mean... C'mon -- Flying Rabbits??? What science fiction principle is this even based on? "Biological gravity wave propulsion"? So now a serum can make cells produce anti-gravity... Yeah, get that mummy tour going ASAP, no miracles here. It's another contrivance meant to pose as evidence that no one at GE was responsible, but it is just farcical when looked at with an analytical eye.
Honestly, with this non-ending parade of contrivances, you have to wonder if the writers put any fore though into the story other than say "Wouldn't it be cool if...?" at that weeks pitch session. You can practically see the series of choices the writers made starting at the core plot in order to justify it unfolding the way they wanted. Since they are writing for a pre-existing series, they failed to realize that changing characters to fit a plot only serves to make the writing seem forced and, well... contrived.