The Sierpinski-Mazurkiewicz Paradox (is really weird)

As a polish person, I respect Zach's refusal to say the name of this paradox haha

Sometimes I feel like the term "paradox" really means "this situation works because of a concept I don't fully understand yet"

It really is fascinating the difference in tone between your videos explaining concepts here and the just raw madness of your humor on your personal channel. I also feel like Zach Star Himself would give Zach Star a wedgie.

Putting the sponsorship before the solution so that we'd have more time to try the problem was real sneaky. Zach is so good at getting us to not fast forward through those :D

It's really hard to focus on the science stuff after binge watching "zach star himself" :D

That's wild. Any ostensibly geometric problem whose solution comes from algebraic geometry.. count me out.. It's mind bending stuff.

(before watching the answer) I found a solution for B’ = S. Since we deal with rotations, let’s start with a unit circle U centred at the origin. Let’s start with A = {(1,0)} and B = U\A. Now B is a circle with a hole on the right. If we rotate B by 1 radian, the hole will land on a point 1 radian along the circle (p1), which is not in B’, but is in S. So let’s remove p1 from S. Now B and B’ have two holes and p1 doesn’t appear in either. However the new hole now lines up with a point 2 radians along (p2), which is in S, but not in B’. So let’s remove p2 from S. Continuing this logic we will remove point that are n radians along the circle (p1, p2, …) from U. So A = {p0} = {(1,0)} and B = U\{p0, p1, p2, …}. Now every hole in B’ is p1, p2, … and as such B’ = U\{p1, p2, …} which is exactly S. p0 is not removed, i.e. not in the list p1, p2, …, since if it corresponded to a point pn, this would imply n=2πm, and thus π=n/(2m) making π rational, which it is not. I’m sure this can be extended by adding all point +1 along the x-axis, rotating and removing point, and so on.

I am so happy this was suggested to me, thank you so much for making this. It may seem a bit silly, but I found a real passion for mathematics in university and examples like this were my driving motivation for learning. I have since moved into development work, and been promoted into a managerial role where I work on interesting things, but have lost touch with this, lovely, interesting and beautiful side of mind play. This felt like discovering your favourite album growing up had an extra track you just hadn't heard, and it being exactly as rewarding as you recall it used to be. Thank you

Few are capable of reading such a title and not clicking instantly.

This paradox is basicly just Hilbert's Hotel all over again, right? You take an infinite number of objects, apply a specific transformation to all of them, and you end up with all the objects you started with, plus infinetely more new ones.

A part of the set is actually nice to visualize. Just imagine the positiv integers wrapped around the complex unit circle A supset of the Projektion points has this magnificent subset rotational property

I appreciate the paradox, although whenever something unintuitive/apparently contradictory happens because we used an infinite set, it sorts of feels like cheating =P

I was waiting for this kind of videos of yours for months. I love your work as a mathematical content creator. U re one of the bests.

Is that really a paradox? It would be a paradox if a set was equal to one of its proper subsets, but with shifting or rotating that should not be impossible. It is like the hotel with an infinite number of rooms, where all rooms are occupied, moving an infinite number of guests to new rooms creates room for new guests. Moving the guests is the key.

The essence of paradoxes, which I've distilled after studying a large number of them, is that there are 2 types: 1. Arising from a negated self-reference. In this case, the proposition you started with is false. It's just a subset of proof-by-contradiction. 2. Involving infinity, in which case, it's not really a paradox; it's just counterintuitive to those not versed in the magic of infinities.

Wow, great video! And I love the proof-sketch style, extremely clear. I understood all of this perfectly (minus the fact that I'm forgetting how to do rotations in the complex plane, but I remember having done those). Thanks !!!!

I would still be cool to see an animation of the points on the complex plane moving around. I know theres infinite of them, but just the ones that fit on screen :P Because I'm still trying to wrap my mind around what the rotation looks like on these points.

I'm pretty sure the proposed set S is dense in the plane (haven't proved it yet). The building blocks required for that proof would be something like this: 1. The set of powers of p is dense in the unit circle. Therefore, you can approximate any direction with arbitrary precision. 2. You can make points arbitrarily close to the origin (for some n,m such that p^n and p^m have almost opposite direction, in which case p^n+p^m is "almost" 0, has very small size). 3. You can multiply p^k by integers so you can get a rough approximation of any point in the plane (for example, choose the floor or ceiling of the size of the target point). 4. Add to that rough approximation appropriate small numbers of the form p^n+p^m in order to get an arbitrarily good approximation.

Great to see a new math video! I always learn something new

Very clever in that it contains a traditional translation encoded by a shift of one to the left that drops all of the constant terms by one and a sort of “power translation” encoded in the rotation that drops all of the powers of each term of the polynomial by one. I could imagine this perhaps being discovered in reverse, though. That is, someone studying sets that do the aforementioned types of transformations along with their associated operations and then realizing they could form the weird “paradox” presented.