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### Quantum Test Found For Mathematical Undecidability

You might counter that if both balls of mud have the same mass (i.e. 1 kg), then the total will have 2 kg of weight. Fine. Then I can point you to the Banach Tarski paradox ( http://en.wikipedia.org/wiki/Banach_Tarski_paradox ) which shows that it should be possible to cut a two kilogram ball into finite number of non-overlapping pieces and put together to give two two kilogram balls, so 2=2+2.

You might counter that you can't divide a real world solid the way you can divide a mathematical solid. But in that case, you've shown that the real world is not 100% mathematical in every sense, so all the free variable are interchangeable without consequence.

That's not what you've shown at all. If physics is to be believed then the balls can't be divided past the level of elementary particles, so the "measure" (i.e. the mass) of any real-world object such as these balls is always a well-defined, existing quantity. We've assigned this measure to the real world, and real objects in it such as these balls are always measurable because they are finite unions of elementary particles.

The Banach-Tarski paradox, on the other hand, uses the fact that there exist non-measurable subsets of R^n with the Lebesgue measure. It's a completely different measure than the one we're using in this real world analogy, so it doesn't make the real world any less mathematical because it's not supposed to describe the real world. The real world is still 100% mathematically consistent when you apply the right laws to it -- if I developed a mathematical theory around the law F=ma^2 and then noticed it wasn't like that in the real world, would it make the real world inconsistent or would it mean I'm using the wrong mathematical framework?

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