0
0

Thoughts on the Banach-Tarski Paradox (self.math)

Phatdave14 が 投稿

So I watched Vsauce's video on the banach-tarski paradox and the part where he talked about the circle missing all the polar points.

I'm only in highschool so correct me if I'm wrong

Points technically don't have an area. The are infinitely small, and within every point there is an infinite more amount of points. So in theory by taking apart this sphere into different sets of points, do they really exist? If they have no area, what is there to reproduce?

That brings me back to where he mentions how the missing points would fill in the same way that dudes hotel would. But within each missing point, since points are infinitely small, they should contain infinite missing points. What they created using banach-tarskis paradox are spheres that are there, but not really there.

Now this is where things get weird. You now have two objects that have zero area. They exist but they don't exist. It doesn't follow general laws of physics. You know they're there mathematically but you can't measure them. Much like dark matter, scientists can tell it is there but so far it can't be measured .

My main question coming out of this is could banach-tarskis paradox be somehow linked to dark matter? I just thought about this while I was in the shower, correct me where I'm wrong and possibly enlighten me a little more on what's known about dark matter, just wanted to share my thoughts in hopes of someone maybe helping me sort through them

全 12 件のコメント
並べ替え:
ベスト
トップ新着論争中古い順Q&A

[–]skaldskaparmal 15ポイント16ポイント  (2子コメント)

Points technically don't have an area.

To be even more technical, they have area 0.

within every point there is an infinite more amount of points.

This is not correct. A point is a single point.

So in theory by taking apart this sphere into different sets of points, do they really exist?

What do you mean by really exist? This is an abstract mathematical framework, not the real world. In the precise mathematical sense of existing, yes, points exist.

If they have no area, what is there to reproduce?

The point. The point is a mathematical object that can be manipulated with mathematical operations. For example, the point in three dimensions, (1, 2, 3) can be translated by 4 units in the x direction, which results in the point (5, 2, 3). I have just done an operation to a point. Doing similar but more complicated operations to other points results in the paradox.

But within each missing point, since points are infinitely small, they should contain infinite missing points.

Again, they do not.

What they created using banach-tarskis paradox are spheres that are there, but not really there.

"Really there" is not a mathematical property. A sphere is made up of infinitely many points, and we can translate the points and rotate them about axes and in the end we end up with two spheres of points, where each point in both new spheres is accounted for.

It doesn't follow general laws of physics.

This has nothing to do with physics. It's purely a mathematical theorem.

My main question coming out of this is could banach-tarskis paradox be somehow linked to dark matter?

Dark matter is a scientific hypothesis to account for a bunch of gravity that we observe that is not accounted for by regular matter. We call it dark matter because we can't interact with any such matter with the electromagnetic spectrum. It has nothing to do with Banach Tarski, which is a mathematical theorem.

[–]FST 4ポイント5ポイント  (0子コメント)

So in theory by taking apart this sphere into different sets of points, do they really exist?

What do you mean by really exist? This is an abstract mathematical framework, not the real world. In the precise mathematical sense of existing, yes, points exist.

A different interpretation of what OP asked, whether the sets exist, is actually a very deep question. There are models of ZF where the four core sets (or whatever they're called -- it's been a while) of the Banach-Tarski construction don't exist. To prove that they exist, we need to add in another axiom, usually the Axiom of Choice or the Ultrafilter Lemma.

[–]Phatdave14[S] 0ポイント1ポイント  (0子コメント)

Thank you for clearing that up for me

[–]zifyoip 4ポイント5ポイント  (0子コメント)

within every point there is an infinite more amount of points

...

But within each missing point, since points are infinitely small, they should contain infinite missing points.

No, this isn't true. A point does not contain other points inside it. A point is a point. Euclid defined a point as "that which has no part."

You now have two objects that have zero area. They exist but they don't exist.

Zero area does not mean nonexistence.

It doesn't follow general laws of physics.

That's true, the Banach–Tarski paradox does not follow the laws of physics. It is not a statement about the physical universe—it is a purely mathematical statement.

You know they're there mathematically but you can't measure them.

Zero area also does not mean the area cannot be measured. The area can be measured—it's zero.

My main question coming out of this is could banach-tarskis paradox be somehow linked to dark matter?

No.

[–]LawOfExcludedMiddleCryptography 2ポイント3ポイント  (0子コメント)

Also, note that infinitely complex spheres have never been observed to exist in nature, thus this is purely a mathematical thing and has no current physical implications.

[–]LawOfExcludedMiddleCryptography 1ポイント2ポイント  (6子コメント)

Alrighty, lemme help you out here.

Points technically don't have an area. The are infinitely small, and within every point there is an infinite more amount of points. So in theory by taking apart this sphere into different sets of points, do they really exist? If they have no area, what is there to reproduce?

A single point has no area, correct; but an infinite number of points do. For instance, a line is merely an infinite number of points in some two directions, and a line segment is an infinite number of points between two points. Likewise, a square is also composed of an infinite number of points.

But within each missing point, since points are infinitely small, they should contain infinite missing points. What they created using banach-tarskis paradox are spheres that are there, but not really there.

No, they definitely made spheres. Note that a point is infintesimally small, and, as such cannot "contain" anything else.

Now this is where things get weird. You now have two objects that have zero area. They exist but they don't exist. It doesn't follow general laws of physics. You know they're there mathematically but you can't measure them. Much like dark matter, scientists can tell it is there but so far it can't be measured .

But it exists (assuming modern theories are correct; unless you believe in the whole [;F=\frac{GMm}{r^2}+\alpha_0;] thing, but gravitational lensing can be used to debunk that). I suggest that you actually take some time to read up on dark matter, because you don't seem to understand it. I can't come close to typing anything out now, but don't believe documentaries with MIchio Kaku too much. But the fact that we have no way to measure it does not mean that it "both exists and doesn't exist", it just means that we have no way to measure it. If you could not figure out how to weigh an apple, would the apple not exist?

[–]zifyoip 2ポイント3ポイント  (3子コメント)

A single point has no area, correct; but an infinite number of points do.

Well, an infinite number of points may. Not all infinite sets of points have nonzero area.

[–]LawOfExcludedMiddleCryptography 0ポイント1ポイント  (2子コメント)

Not all infinite sets of points have nonzero area.

Example?

[–]zifyoip 4ポイント5ポイント  (1子コメント)

A line has zero area.

The set of points {(0, 0), (1, 0), (2, 0), ...} has zero area. More generally, any countable set of points has zero area. So Q2, the set of all points with rational coordinates, has zero area.

The Cantor set has zero length. Therefore, the set C × R, where C is the Cantor set, has zero area.

The Sierpinski carpet has zero area.

[–]LawOfExcludedMiddleCryptography 1ポイント2ポイント  (0子コメント)

Honestly, Wikipedia's a pretty good resource: https://en.wikipedia.org/wiki/Dark_matter#Baryonic_and_nonbaryonic_dark_matter, /u/PhatDave14.

[–]Phatdave14[S] 0ポイント1ポイント  (0子コメント)

Thanks for helping clear that up :)