I Physical implications from Vitali sets or Banach-Tarski?

[greypilgrim]

Hi.

Can we infer something about physics from stuff like Vitali sets or the Banach-Tarski paradox? Maybe if we assume the energy in a given space volume to be well defined and finite, that there must be fundamental particles that can't be split, or that there must be a Planck length and energy or something along those lines?

 

[mathman]

The physics and mathematics you are describing are completely unrelated.

 

[greypilgrim]

How so? I need a measure to quantify the energy in a space volume. And that measure better not allow me to double the energy just by decomposing and rearranging the objects in it.

 

[Samy_A]

How so? I need a measure to quantify the energy in a space volume. And that measure better not allow me to double the energy just by decomposing and rearranging the objects in it.

The sets involved in Banach-Tarski, as well as the Vitali sets, are not measurable.
As @mathman said, no relation between these sets and physics.

 

[greypilgrim]

Well that's basically my question. Can we conclude from this that non-measurable sets can't exist in physics, because they would violate conservation laws?

 

[Samy_A]

Well that's basically my question. Can we conclude from this that non-measurable sets can't exist in physics, because they would violate conservation laws?

I think so, but better wait for a physicist to give a reliable answer.

 

[TeethWhitener]

Can we conclude from this that non-measurable sets can't exist in physics, because they would violate conservation laws?

I think the point is more that in order to get a conservation law, you're probably going to have to do an integration at some point (it's either that or go the differential route and wrestle with issues of continuity/differentiability), and you can't integrate without a measure. So in that regard, maybe non-measurable sets exist but are functionally useless (at least in physics).

 

[mathman]

What is a physical set? If it is something made of stuff, it is finite and always measurable.

 

[greypilgrim]

What is a physical set? If it is something made of stuff, it is finite and always measurable.

Only if you assume the existence of elementary particles. If you don't, you could theoretically split something infinitely and maybe rearrange it like Banach and Tarski did. That's basically what I wondered in the original post, if we can conclude from the fact that we don't observe this, that there's only a finite number of stuff, or energy.

I think the point is more that in order to get a conservation law, you're probably going to have to do an integration at some point (it's either that or go the differential route and wrestle with issues of continuity/differentiability), and you can't integrate without a measure. So in that regard, maybe non-measurable sets exist but are functionally useless (at least in physics).

That raises the question if there are physical processes that can transform non-measurable sets (in phase space, I guess) into measurable sets and vice versa. I have no idea if such a dynamics is possible and how it could look like...

 

[Samy_A]

Only if you assume the existence of elementary particles. If you don't, you could theoretically split something infinitely and maybe rearrange it like Banach and Tarski did. That's basically what I wondered in the original post, if we can conclude from the fact that we don't observe this, that there's only a finite number of stuff, or energy.

Even in mathematics we don't construct these non measurable sets. We invoke the axiom of choice to deduce their existence.
But sure, if you change the current paradigms of physics, and if you postulate the existence of a method to construct in a lab something (a non measurable set) that even in abstract mathematics isn't constructed, then yes, you could "theoretically" (no idea in what theoretical context), do anything.

But if you accept the laws of physics as they are known, the answer is still no: you can't become rich by repeatedly applying Banach-Tarski to an ounce of gold.

 

[TeethWhitener]

Even in mathematics we don't construct these non measurable sets.

I'm not an expert in this area, but is this true? What about Vitali sets? Are they not constructed?

 

[Samy_A]

I'm not an expert in this area, but is this true? What about Vitali sets? Are they not constructed?

At some point one invokes the axiom of choice, at least in the version I know.

See here, for example, or on Wikipedia.

Solovay's theorem shows that without the axiom of choice, but with an added condition to ZF, all real sets are Lebesgue measurable.

 

[WWGD]

Well that's basically my question. Can we conclude from this that non-measurable sets can't exist in physics, because they would violate conservation laws?

AFAIK, you need to have an uncountable number of particles/atoms in order to make the construction. I guess this would constitute a physical limitation.

 

[greypilgrim]

AFAIK, you need to have an uncountable number of particles/atoms in order to make the construction. I guess this would constitute a physical limitation.

My question from the original post was actually addressing this, but in the other direction: Could nature's "mechanism" to forbid such constructions be that it limits the number of particles to a finite number? Or that it limits the number of energy chunks by exhibiting a smallest possible energy quantum, the Planck energy?

 

[WWGD]

My question from the original post was actually addressing this, but in the other direction: Could nature's "mechanism" to forbid such constructions be that it limits the number of particles to a finite number? Or that it limits the number of energy chunks by exhibiting a smallest possible energy quantum, the Planck energy?

Good question, I will give it a thought, but my knowledge of Physics is limited.