Strange/Useless powers.

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The power to fly, but only on your nose

Generated on my überpowered Xperia Z2
 
I have that power. And yes, I do know about the barber pair of docks. I also know the resolutions to it.

The whole realization that you could describe things that were not sets, and the whole need to verify that what looked like a set was a set was big news to mathematicians / logicians, as well as the general "don't define a set by negative parameters". Basically, unless you know the whole universe of objects, a "what's not in" is not well defined.

Modern set theory and axioms are ... confusing, at best. I have read the sections on Wikipedia several times, and I still don't get it. But I also know that it's only one of several equivalent systems, and not all of them require an infinite set of axioms to define sets.

Finite universes are trivial. But those can't describe math. It's the whole Godel issue.
 
Well in a finite universe N doesn't exist. In the ZFC system its existance is guaranteed by the axoim of infinity. But then again basically every theory using N is either incomplete (i.e. there exists a statement A where you can't prove that A is true nor that it is false) or contradictory, courtesy of Gödel. As a result, we can't prove that our system of axioms is contradiction-free using only the axioms themselves, which kinda makes sense.
 
So close. I can throw it in anyways.

If Schrödinger's cat walks into a forest, and nobody is there to observe it, is it really there?

The power to understand that.
 
The power to remove powers from oneself

Sent from my Xperia Z2 running smooth Android L.
 
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