Banach-Tarski Paradox. First stated in , the Banach-Tarski paradox states that it is possible to decompose a ball into six pieces which can be reassembled by rigid motions to form two balls of the same size as the original. In Banach-Tarski paradox, you are given the power to pick up infinitely many points at once, but you can only perform 'rigid motion' with them, like translate them. The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original Stefan Banach · Alfred Tarski · Tarski's circle-squaring problem.
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First of all, if we didn't restrict ourselves to rigid motions, this paradox would be more believable.
For instance, you can take the interval [0,1], stretch it to twice its length and cut it banach tarski paradox 2 pieces each the same as the original interval. A second caveat is that banach tarski paradox actual semantic content of this post is going to be extremely low.
Nevertheless I believe this viewpoint is somewhat clarifying as to the nature of these paradoxes, and as to how they are not as fundamentally tied to the nature of sets or the nature of infinity as one might first expect.
A three-dimensional Euclidean ball is equidecomposable with two copies of itself.
Banach–Tarski paradox - Wikipedia
In fact, there is a sharp result in this case, due to Raphael M. The strong version of the paradox claims: Any two bounded subsets of 3-dimensional Euclidean space with banach tarski paradox empty interiors are equidecomposable. While apparently more general, this banach tarski paradox is derived in a simple way from the doubling of a ball by using a generalization of the Bernstein—Schroeder theorem due to Banach that implies that if A is equidecomposable with a subset of B and B is equidecomposable with a subset of A, then A and B are equidecomposable.
The Banach—Tarski paradox can be put in context by pointing out that for two sets in the strong form of the paradox, there is always a bijective function that can map the points in one shape into the other in banach tarski paradox one-to-one fashion.
The proper formulation of the intuitive idea that probability always makes sense, is that every subset of R is Lebesgue measurable. This is consistent with ZF with dependent countable choice, and the axiom of banach tarski paradox, but not with the axiom of choice as applied to all the real numbers.
Then the Banach Tarski thing doesn't work. The mathematical version of the paradox uses the concept banach tarski paradox an immeasurable set.
Banach-Tarski paradox | What's new
Every object in real life is measurable, because it is the set of a finite number of atoms taking up a finite amount of space. Mathematically, even when finite banach tarski paradox infinite, you still usually have measurable sets.
You really have to try very hard in order to create an immeasurable set. The Banach-Tarski paradox splits the sphere into a finite number of immeasurable sets of points.
I agree that "one could easily imagine performing set operations in a universe where there is no binding energy", but I think that one could imagine this in many different ways. Since the Banach—Tarski theorem banach tarski paradox rather subtle, I think we banach tarski paradox admit the possibility that its bearing if any on such an imaginary universe might depend on the details of this imaginary universe.