Problem with Axiomatic Systems

It was a cold evening in Hell. Our friends found themselves barely able to look at their algebra books any longer. Everything was becoming a crazy jumble of gibberish that only the most skilled could make into words. They were exhausted by the volume of work that had been required of them by their leader Mij.

"HE IS THE DEVIL," someone exclaimed!

Everyone nodded in agreement. "I bet we could prove it," said another. Thus our heroes had a new mission.  

After some work they came up with the most elegant proof. Proud of their work, they displayed the proof on the board for Mij to discover the next day. 

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Mij is Satan if and only if Modern Algebra is Hell.

Assume Mij is Satan.  We know that Mij is the ruler of Modern Algebra. Since Mij is Satan, Satan is the ruler of Modern Algebra. Since Satan is the ruler of Hell, Modern Algebra must be Hell.

Assume Modern Algebra is Hell. We know Satan is the ruler of Hell, so Satan must also be the ruler of Modern Algebra. Since, Mij is the ruler of Modern Algebra, it is trivial that Mij is Satan. 

QED

By Inspection, Modern Algebra IS HELL! Therefore, it is is clear that Mij is Satan.

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Mij walks into the classroom the next morning as the class holds back their giggles. Surely, he will be disgusted, or secretly amused, and will shake his head as he has to erase this obvious display of insolence.

Mij stands back and reads the proof providing no reaction to our friends. Their disappointment takes over as Mij begins to erase the board.

The only thing Mij says is, "the problem with faulty axiomatic systems is that you can prove anything."

Like a demonic superhero Mij both teaches a valuable lesson and stomps on their good time in one fell swoop.  He really is the Devil.