Naked Science Forum
General Science => General Science => Topic started by: talanum1 on 06/06/2022 12:12:26
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It is: we have the Banach-Tarski Paradox which says that one sphere is two spheres or 1 = 2 or 1 ~= 1. One can also prove 1 = 1. So here is a inconsistency!
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Not really. The BT paradox is only paradoxical if you use two different definitions of volume.
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Hi.
The Banach-Tarski paradox relies on the Axiom of Choice (well technically something slightly weaker than the full AC but certainly stronger than just ZF). Anyway, the important thing is that in some set theories, you can't make two spheres from one. Pure Mathematicians often do try to avoid invoking the Axiom of Choice and certainly do identify where it has been used (because they know it's controversial and someone may want to prove a result without assuming the Axiom of Choice).
There's nothing inconsistent about constructing two spheres from one sphere. It's unexpected and counter-intuitive but it would only be inconsistent if you showed it could be done and also it could not be done. So it only gets the name "paradox" because it's counter-intuitive. However, in its usual form it involves something that cannot be done in practice, it is only a mathematical consequences of a hypothetical cut. The way you have to cut up the original sphere involves either (i) chopping it up into an infinite set of pieces, or else (ii) chopping it into a finite set of extremely pathologically shaped pieces where the cut would never be completed (e.g. some cuts zig-zag through the sphere so much that the total length of the cut is inifnite).
There never was any guarantee that our common life experience would hold for infinite cuts like this.
Best Wishes.
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Is Mathematics Inconsistent?
The question is inconsistent, since it assumes that there is one kind of mathematics.
The mathematics of primary-school integer numbers is consistent, provided you declare a few things illegal, such as "dividing by zero", or "subtracting a larger number from a smaller number". The Banach-Tarski Paradox never even comes up in this context.
Godel proved mathematically that you can't prove everything mathematically.
- Any sufficiently complex mathematical system is either inconsistent or incomplete
- And you can't prove which it is, within that system
- Primary-school numbers are not sufficiently complex to cause many problems
- It is good that we have multiple mathematical systems that can take on different problems, using different assumptions/axioms
- For example, we have high-school integer numbers, which can cope with "subtracting a larger number from a smaller number"
See: https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems