Paradaxe

It's made out of the same thing its meant to cut down. Plus, it's useless. You can't cut wood with wood.
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Also the handle is metal while the blade is wood when the blade should be metal and the handle should be wood.

That's not a paradox though...a statement or proposition that, despite sound (or apparently sound) reasoning from acceptable premises, leads to a conclusion that seems senseless, logically unacceptable, or self-contradictory.

I think the better name would be "The Irony Axe"