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By that reasoning, no numbers exist at all.

What is the usefulness of real numbers, beyond the usefulness of computable numbers? Real numbers give you silliness like "You can cut and reassemble a ball into 2 balls identical to the original, if you cut it into 5 infinitely detailed pieces. Negative and Zero don't lead to patently impossible predictions about the world.

https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox






You can't do it with a real sphere because it's made of atoms and bends the space around it. Forget spheres, you can't even cut a string into infinitely many pieces.

That's not inherent to the Real numbers, that requires the Axiom of Choice as well. The Reals are well-founded without that.



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