Well, THER'S YER PROBLEM.
You're using math logic. Allow me to give the full version of the story. I was measuring the resistance of different parts of an electric circuit and had just made a pretty shiny graph showing my data in Excel. When the professor walks by we ask him if it looks all right. He said, "well, what happens when you divide by zero?"
"The universe explodes!" said I.
"You can't divide by zero!" said my lab partner.
"This isn't a math class! In physics a division by zero is infinity."
So our diode had infinite resistance at certain voltages according to the graph.
The end!

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