The domain of infinity is strange and unpredictable. Oftentimes. it can lead to results that lies in contrast to what we are used to in our daily lives. As mentioned before, infinity is a theoretical concept that lies purely in our imagination. A very popular result of an infinite domain that it is doesn’t alter in size no matter how many operations we perform on it. In fact, even if we were to separate out an infinite number of points from infinity,(that’s right!), we would’nt be changing it either. Banach Tarski paradox is a natural and interesting consequence of such property.
THE PARADOX AND ITS BASIS
A 3-d solid ball can be decomposed into disjoint subsets which if re-arranged and put together, can form two identical copies the same size of the first 3d ball! It is one of the many bizarre consequences of the infinite domain and what infinity means even if we were to limit ourselves strictly in the mathematical domain. It is paradoxical in the sense, the any sense of conservation found in limited entities vanishes while dealing with points of a geometrical shape.
WHY THIS PARADOX GOES AGAINST COMMON SENSE
No object in real life is perfectly geometric. Perfect shapes are merely fictitious reference bodies based on which real life objects are modeled. We can say that a dvd disk or a frisbee is closer to being a circle than it is a square, but it doesn’t make it a perfect circle. A perfect circle or any other geometric shapes can be accurately described through numbers, more accurately- a range of numbers and graphs and can be resolved into an infinite number of points.
THE INFINITY OF POINTS
The infinite points inside a 2-d shape or 3-d shape is uncountable.The transition between a single point and a smallest line segment one can imagine, is an abrupt jump from unity to infinity. There is nothing in between, (except maybe for a collection of disjoint individual points considered as whole). . We cannot jump from one point to another, in a line segment, without leaving behind an infinite points in between and that serves as an important tool to understand its not-so intuitive nature. Two lines, irrespective of their lengths contains the exact same number of uncountable infinite points. Conversely, it can be said that removing finite number of points from the sea of infinity doesn’t alter its size, not even to the slightest extent. Thus, in removing a point from a line or any 2-d shape for that matter and placing it anywhere else, we have literally created a point out of thin air. A better way of understanding this is when we define a shape within certain boundaries(given in 2-d Euclidian co-ordinate systen), we are essentially saying that an “infinite number of points has arranged itself in such shape in the specified boundaries.”
IMPLICATIONS OF AN INFINITE POINTS(OPTIONAL)
Did you know that a line-segment of any size(irrespective of its length) has the same number of uncountable infinity points?A longer line segment can be formed by simple re-arrangement of the infinite points inside a smaller one and the converse is also true. Say a smaller line segment exists from x = 0 to x=k. We can form a longer line segment of length 2k by stretching it such that it exists from x = 0 to x =2k. We can say the extreme point in position k of the smaller segment has now moved to position 2k after stretching, whereas the points in position 0 coincide with each other. Every point in between has moved to new position in proportion. Thus, the newly formed longer line segment occupies a larger space, yet its points are just as dense as before i.e. points spreading apart doesn’t seem to create any pointless space in between!!(impossible in real life). To conclude, we can remove, a large number of points(even infinite) from a segment and still be left with an endless supply of it. This is because, points by virtue of being dimensionless doesn’t occupy any space and thus can be packed in an infinitely dense manner (as opposed to the limitations of what is considered a very very small particle in our universe viz atomic and sub-atomic which still occupy space and fails to qualify as a point). Thus, for this paradox to apply, the shapes used must be infinitely divisible and have the same resolution as the real number line. (The smallest length to which space can be resolved, is known as the Planck’s length, which is roughly equal to 1.6 x 10–35 m or about 10–20 times the size of a proton. Any measurable phenomenon which is a function of length, when observed within a spatial dimension lesser than equal to this value, is for all practical purposes constant! Thus, reality itself has its own set of limitations which limits the above paradox to be.)
BANACH-TAR SKI PARADOX
A popular You-tuber Michael Stevens, in his channel V Sauce has explained this paradox marvelously in a video whose link has been provided below. It also explores the relatively simple explanation of hyper-Webster infinite dictionary along with a couple other examples, that serve as a precursor to this seemingly complex idea.