Banach-Tarski and the Paradox of Infinite Cloning • Hacker News Board

Re: Banach-Tarski and the Paradox of Infinite Cloning

#141

Post by nwallin » Mon, Aug 30, 2021, 10:38 PM UTC

DerekL wrote: post #123

perl4ever wrote: post #119

Until recently I never questioned the idea that, say, the positive integers and the odd positive integers are equivalent because they can be paired, but this cloning thing seems like something that falls out of that. And it seems like that view of infinity isn't actually necessary if Cantor style cardinality is not the last word. In the paragraph on nonstandard analysis in the Wikipedia page on infinity, it says: "Th…

> Until recently I never questioned the idea that, say, the positive integers and the odd positive integers are equivalent because they can be paired, but this cloning thing seems like something that falls out of that. They really aren’t connected. The first statement (the positive integers can be partitioned into two sets, each of which has the same size as the original set) follows from the usual axioms of set theo…

Isn't that just because Hilbert's Hotel is a property of the natural numbers (well ordered) while Banach-Tarski is a property of the reals? (not well ordered without AoC) We can split the natural numbers into odd and even groups by starting with 1 and iterating on the odds, and starting with 2 and iterating on the evens. But because the reals are not well ordered, the step in Banach-Tarski where we pick an arbitrary point that hasn't already been grouped into a set is impossible.

The natural numbers (and therefore Hilbert's Hotel) provide a natural way to say "whatever, just pick one" but we need to invoke the well-ordering theorem (which is equivalent to the Axiom of Choice) make the same "whatever, just pick one" statement about the reals. (and therefore Banach-Tarski)

Re: Banach-Tarski and the Paradox of Infinite Cloning

#142

Post by lostmsu » Mon, Aug 30, 2021, 10:43 PM UTC

Viliam1234 wrote: post #33

smiley1437 wrote: post #25

Is this basically the same premise as Zeno’s arrow, but with more steps?

An interesting difference is that the Banach-Tarski trick works in 3D, but not in 2D or 1D.

In 1D there are much simpler tricks.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#143

Post by Nevermark » Tue, Aug 31, 2021, 4:04 AM UTC

After reading the approach it seems like too much work! Anyone willing to critique my suggested simpler proof (which didn’t occur to me until after reading the article).

TLDR; It is basically the same as proofs that all countable sets have the same cardinality. (TLDR of that: map set of positive integers x to the even numbers by doubling, and the odd numbers by doubling and subtracting 1. Take the union of even and odd and you end up with the set you started with, the positive integers x).

For a circle:

We can identify all the points on a circle as the points p associated with the [x,y] coordinates of the complex numbers p = e^(2.c.i.pi), where 0 If we take each of those points p and rotate it by doubling its c, we now have the same points represented by the expression p = e^(2.c.i.pi), where x So the same number of points, but two passes around the circle, 0 We have now rearranged points of one circle into two.

For the surface of a sphere:

We simply divide a sphere up into points defined by a stack of circles at real-valued vertical z positions, z Again, rotate the points by doubling c, so that they are now located at c, where 0 Similar generalizations work for including the volume.

Anyone understand why this simpler proof is wrong, or why the more complex proof in the article does something better?

Re: Banach-Tarski and the Paradox of Infinite Cloning

#144

Post by danparsonson » Tue, Aug 31, 2021, 8:44 AM UTC

wizzwizz4 wrote: post #133

danparsonson wrote: post #127

Earlier quoted context omitted.

I'm not sure what you mean - any battery you have would seem to contradict you - would you elaborate?

The electrons are not wholly within the battery; they are mostly within the battery. (Batteries discharge over time for an almost completely unrelated reason.)

I think you're misinterpreting the meaning of the wave equation. Batteries self-discharge due to chemical reactions not quantum tunneling.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#145

Post by ncallaway » Wed, Sep 01, 2021, 9:46 PM UTC

thaumasiotes wrote: post #138

ncallaway wrote: post #134

Earlier quoted context omitted.

Me? I have no idea. I was just trying to catch and diffuse what seemed like a miscommunication between two people. Actually selecting and proposing an axiom set is way outside my knowledge-base. My limited understanding is that the Axiom of Choice, in ZFC leads to Banach-Tarski, and if it’s removed Tarski doesn’t hold, but I don’t have nearly enough information to say if that’s worth exploring removing it.

The Axiom of Choice is one of the most controversial things in mathematics. There are many people who choose to work without it. I'll give a statement of it here: The Cartesian product of non-empty sets is itself non-empty. The Cartesian product of sets S_1, S_2, S_3, ... is of course the set of tuples (s_1, s_2, s_3, ...) such that s_1 ∈ S_1, s_2 ∈ S_2, s_3 ∈ S_3, ... . An element of the Cartesian product is a tuple…

This is a great overview and explanation of the Axiom of Choice! I was familiar with the basics of it, but this really did a good job clarifying it for me, so thanks for that.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#146

Post by x3n0ph3n3 » Thu, Sep 02, 2021, 4:14 PM UTC

Smaug123 wrote: post #66

x3n0ph3n3 wrote: post #65

Earlier quoted context omitted.

One thing worth pointing out is that our universe operates on the integers rather than the real numbers, and the Banach-Tarski requires operating on the reals.

Is that known? It's an appealing idea, but bearing in mind that general relativity is very resistant to quantisation, I'm not sure I'd be comfortable to declare it as fact.

It's implied by quantization. There is a minimum divisibility of space (plank lengths) and objects (fundamental particles).

Re: Banach-Tarski and the Paradox of Infinite Cloning

#147

Post by Smaug123 » Thu, Sep 02, 2021, 9:29 PM UTC

x3n0ph3n3 wrote: post #146

Smaug123 wrote: post #66

Earlier quoted context omitted.

Is that known? It's an appealing idea, but bearing in mind that general relativity is very resistant to quantisation, I'm not sure I'd be comfortable to declare it as fact.

It's implied by quantization. There is a minimum divisibility of space (plank lengths) and objects (fundamental particles).

I specifically said that the resistance of GR to quantisation is why I'm not very happy saying that it's fact.

Re: Banach-Tarski and the Paradox of Infinite Cloning

#148

Post by Smaug123 » Thu, Sep 02, 2021, 9:37 PM UTC

Nevermark wrote: post #143

After reading the approach it seems like too much work! Anyone willing to critique my suggested simpler proof (which didn’t occur to me until after reading the article). TLDR; It is basically the same as proofs that all countable sets have the same cardinality. (TLDR of that: map set of positive integers x to the even numbers by doubling, and the odd numbers by doubling and subtracting 1. Take the union of even and o…

I didn't really read your proof, because it's late and I'm tired, but there is no Banach-Tarski construction in 1 or 2 dimensions (whether or not you like the axiom of choice). The third dimension is crucial. So if your proof doesn't somehow intrinsically rely on "n >= 3", it can't be right.