Imagine two friends hiking through the woods. They become hungry and decide to split an Apple, but half of an apple feels too small. One of them recalls the most bizarre idea she's ever heard. It is a mathematical theorem that involves infinity, which makes it possible to transform one apple into two.
This argument is known as the Banach-Tarski paradox after Alfred Tarski and Stefan Banach, two mathematicians who created it in 1924. It shows that it is possible to break a three-dimensional solid into pieces that can recombine to create two identical copies of its original. Two apples for the price of one.
It is immediately apparent that it is counterintuitive," said Dima Sinapova, University of Illinois, Chicago.
The paradox results from the most complex concept in math, infinity.
Infinity is a number that feels like it, but it doesn't behave like one. Any finite number can be added or subtracted to infinity, and the result will still be the same infinity as the one you started with. However, not all infinites are created equal.
Mathematicians have proven that some numbers are larger than others over the past century. The natural numbers 1, 2, 3 and so forth are examples of a countable infinite. Although they can go on forever, it is possible to count them off (e.g. listing the numbers 1 through 1 Trillion).
Contrary, the real numbers, all the infinitely numerous tick marks that denote decimals, are an infinity. It is impossible to count all real numbers that lie within any interval of the number line. Even a small one like the one between zero and 1.
This distinction between countable infinities and uncountable infinities causes the natural numbers to be smaller than the real numbers. Mathematicians refer to this difference by saying that the two have different cardinality.
In 1891 Georg Cantor demonstrated that distinguishing cardinalities requires more than conceptual Jiujitsu. Cantor also demonstrated that an infinite number points on a line have the same cardinality than the infinite numbers of points that fill a volume, such as a sphere.
Banach and Tarski discovered that one sphere can be divided into two. This is done by dividing the uncountably endless set of points it contains into an uncountably infinite amount of countably infinite sets. A very precise dissection process is used to separate the spheres.
Pick a starting point to build one of these infinitely many sets. Any point within the sphere is sufficient. Next, pick two angles that are an irrational amount of degrees. This means that any number of degrees like pi that cannot be written in fractions can be used. You will soon be able to rotate the sphere. These angles are for North-South and East-West rotations, respectively.
Now rotate the sphere North-South, East-West by the appropriate number. The sphere will now land at a new point. This is the second point of your set. The first point is your starting place.
Next, turn the sphere in any of the four directions. The only restriction is that you cannot go backwards in the direction you came. A third point will be created. You can repeat the process infinitely many times to create a set of infinite points.
The set will possess a few key properties. It will not include the same point twice. This is due to the fact that the angles of rotation are irrational. The second aspect is that the set can be countably infinite. You could assign a natural quantity to each point chosen through the rotation process.
Spencer Unger, University of Toronto set theorist, stated that the whole sphere is an uncountable object. It is broken down into many countable pieces.
You can repeat the same process starting at any point on your sphere. Each point is unique and generates its own set of points. This allows you to create uncountably many sets with an infinite number of points.
Once you have the sets, you can sort them into a few groups. The last rotation before landing on a point will identify four groups. The center point of the sphere will be included in the fifth group, as well as all points at the poles. A sixth group will include every point of origin.
These groups could not be combined to produce the spheres that Tarski and Banach wanted. They used Felix Hausdorff's idea to double the number of points. This allowed them to rotate the points in a single group and create a new set of points that was larger than the one they were starting with.
Consider, for instance, the group that includes all points derived by a final rotation to East. Rotate this group to the West. This will instantly negate all the final East rotations, and transform the group into the (still infinite) collection of points that preceded the creation of the original sets during set-building. This group now has points that have been completed on the North, South, and most importantly, East rotations. These were the original foundation of the group. The rotated piece is both new stuff and its original self.
This seems possible because of the nature of infinity. All East rotations are erased if the whole East group is turned West. What then?
Because backtracking was not allowed, there were no West-ending routes that preceded the final Easts. There were many East-ending routes that preceded the final East turns. However, there is no rule against making East-East your two last rotations. There were many paths that ended with Norths or Souths. We have turned the entire East group west, and it now contains all points East, North, and South. All the beginning points are there, because every single East turn path has returned to its source.
We have now duplicated all points in three of six groups (North and South, starting points). The next step is to duplicate the three other groups (East/West and poles/center). These are the easiest: Simply rotate the North group South to get all North, East, and West points.
We must then duplicate the center point and the poles. This is similar to the Hilberts hotel argument, which was created by David Hilbert in 1924. It's also how Banach and Tarski discovered their paradox.
This thought experiment will allow you to imagine a hotel that has infinitely many rooms. Imagine that Room 43 is empty. Move every guest from Room 44 or above down one room. You have filled the vacancy and not created a new one.
Now, imagine the missing poles as vacancies on different lines of latitude. Now, shift all points along each line of latitude to the right and infinity will fill the vacancies.
The empty center point can be filled with the same method on another circle. Voila! All six groups have been duplicated. Each of the six groups can be combined into its own sphere.
It feels impossible. It seems impossible to double an object's volume by simply decomposing it and rearranging. Banach-Tarski helps to take the weight off of uncountability. The sphere can be deconstructed using sequential rotations like counting natural numbers. This creates a smaller workspace and a more manageable infinite than the one that plagued the original sphere.
There are many who see the paradox as a problem. Others see it as an absurd conclusion, which points to a flaw within the rules of mathematical reasoning that allow it.
It's like a bellwether," Norman Wildberger, a retired professor of mathematics at The University of New South Wales, Sydney, Australia, stated. It's like a huge red flag.
The axioms of choice are the mathematical rules that make the Banach-Tarski paradox possible. It is one of nine axioms found in the Zermelo-Fraenkel set theoretical, or ZFC. This theory serves as the foundation for modern mathematics.
The axioms of choice have been criticized for being an addition to the eight. This makes ZFC vulnerable to criticism, especially when they enable outcomes such as the Banach-Tarski paradox. Mathematicians have the ability to select an item from any bin in a collection. Banach-Tarski is a Rorschach test to work with infinitesimal: Many people find the paradox fascinating; Wildberger, on the other hand, cringes.
However, most mathematicians don't lose sleep over the axiom "of choice". The Banach-Tarski paradox is seen by many mathematicians as an example of the richness and beauty of mathematics. It is a law-abiding example showing how math can deviate from physical intuition without contradicting its own.
Sinapova said that almost anything can be multiplied and can be broken down into two items of the same cardinality.
