This fact on 2-dimensional space blows my mind

[Anixx]

This fact blew my mind when I realized it: in 2-dimensional space one can make a bounded set into its strict subset by means of rotation.

I still cannot comprehend how coud this be possible and how much of math gets blown by it. I think, it is used to prove the Tarski paradox. While most mathematitians consider the weakest brick in the building of Tarski paradix the Axiom of Choice, to me it seems that the mentioned above fact is the real cause of the magic of the paradox.

 

[fresh_42]

I remember a colloquium in which the proof of Banach-Tarski was presented, and the lecturer noted that, in his opinion, it was less the axiom of choice but rather our incomplete understanding of the concept of points that causes it. I think that he was right and further evidence is Peano-, Sierpinksi-, Hilbert-, or other space-filling curves. A point is an object that exists although it has no length in any direction. The axiom of choice might be the logical tipping point, but it's the existence of points without any extensions that is playing tricks on us. We get an extension of things without extension if we just collect enough of them.

 

[martinbn]

I think it is important to stop picturing the set in question as a nice blob you can draw. You should think of it as a dust of points in the plane, and it has to be non-measurable.

 

[Anixx]

I think it is important to stop picturing the set in question as a nice blob you can draw. You should think of it as a dust of points in the plane, and it has to be non-measurable.

Non-measurability is not required (its Lebesgue measure is zero) and it can be absolutely fit into the boundary of the unit circle. In other words, it fits in a curcular line.

 

[martinbn]

Non-measurability is not required (its Lebesgue measure is zero) and it can be absolutely fit into the boundary of the unit circle. In other words, it fits in a curcular line.

If you consider the usual measure on the circle, the set will have to be non-measurable with respect to that measure.

 

[Anixx]

If you consider the usual measure on the circle, the set will have to be non-measurable with respect to that measure.

Hmm. Why? The set can be countable, so its Lebesgue measure is 0.

 

[martinbn]

Hmm. Why? The set can be countable, so its Lebesgue measure is 0.

Ok, then how do you prove the statement of the first post?

 

[Anixx]

Ok, then how do you prove the statement of the first post?

Take a point on a unit circle with coordinates $(1,0)$. Now rotate it in the plane counter-clockwise around the point $(0,0)$ by an angle of irrational number of degrees (not radians) $\phi \notin \mathbb{Q}$.

Consider a set of points formed by infinite number of such consecutive rotations $(0\phi, 1\phi, 2\phi, 3\phi,\dots)$ . Now take from this set the initial point $(1,0)$. We obtain a set, which is a strict subset of the original set, yet equal to it via the rotation by angle $\phi$.

 

[PeroK]

This is a variation of the mapping on the set $\{1, \frac 1 2, \frac 1 3, \frac 1 4 \dots \}$ given by $\frac 1 n \to \frac 1 {n+1}$. It's really just a one-to-one mapping of a countable set into itself.

 

[martinbn]

Take a point on a unit circle with coordinates $(1,0)$. Now rotate it in the plane counter-clockwise around the point $(0,0)$ by an angle of irrational number of degrees (not radians) $\phi \notin \mathbb{Q}$.

Consider a set of points formed by infinite number of such consecutive rotations $(0\phi, 1\phi, 2\phi, 3\phi,\dots)$ . Now take from this set the initial point $(1,0)$. We obtain a set, which is a strict subset of the original set, yet equal to it via the rotation by angle $\phi$.

I see, it has nothing to do with Banach Tarski. It is just a wraped around version Hilbert's hotel.

 

[Anixx]

I see, it has nothing to do with Banach Tarski. It is just a wraped around version Hilbert's hotel.

Banach-Tarski uses a similar technique: obtaining a set by rotating a point on sphere by irrational number of degrees.

 

[fresh_42]

I see, it has nothing to do with Banach Tarski. It is just a wraped around version Hilbert's hotel.

Hilbert's rotunda.

 

[Anixx]

This is a variation of the mapping on the set $\{1, \frac 1 2, \frac 1 3, \frac 1 4 \dots \}$ given by $\frac 1 n \to \frac 1 {n+1}$. It's really just a one-to-one mapping of a countable set into itself.

You cannot make $\frac 1 n \to \frac 1 {n+1}$ by moving it.

 

[PeroK]

You cannot make $\frac 1 n \to \frac 1 {n+1}$ by moving it.

None of these countable examples rely on the axiom of choice. The Banach-Tarski paradox goes much deeper. The axiom of choice is a decision about what sort of mathematics we want to study. If we reject the AC, then we can do mathematics differently and there is no Banach-Tarski paradox.

 

[Anixx]

None of these countable examples rely on the axiom of choice. The Banach-Tarski paradox goes much deeper. The axiom of choice is a decision about what sort of mathematics we want to study. If we reject the AC, then we can do mathematics differently and there is no Banach-Tarski paradox.

The AoC is only needed to make an already existing countable paradox into uncountable version.

 

[PeroK]

The AoC is only needed to make an already existing countable aradox into uncountable version.

It's much deeper than that. The AC is needed to prove that every vector space has a basis, for example. It appears throughout mathematics.

 

[martinbn]

The AoC is only needed to make an already existing countable paradox into uncountable version.

Banach Tarski is more subtle. For instance there is no such paradox in the plane. The group of motion in the three dimensional space is important.

 

[statdad]

Picturing things in the plane has been a bugaboo (I've been dying to use that word) for a long time. The prof I had for my first measure theory class told this story for us to chew on.

A physics professor, an engineering professor, and a mathematics professor had all died. They were being tested to see whether they would go to heaven or hell. The test was this:
They would be placed in a large room with a small amount of paint (I don't remember the unit he used, so say a pint). If they could make that paint cover one wall of the room in two hours they could go to heaven, otherwise not.

The first tested was the physicist. When the two hours was done he had painted some, but not all, of the wall, and so was condemned to hell.

The engineering professor was next, and the outcome was the same.

The mathematics professor was next. A little over an hour into the test the professor wandered out of the room, found the proctor, and said "I'm done."

They walked into the room and the proctor was astounded to see that the wall was covered.

"You're the first to ever succeed, how did you do it?" the professor was asked.

"Simple. I placed the origin in the corner and only painted the points with rational coordinates."