Counterintuitive and bewildering, but true: there exist as many even numbers as there exist natural numbers, for both are infinite.
However, there are not as many real numbers as natural numbers; in fact, there are infinitely more real numbers than there are natural numbers. Why is the infinity of real numbers greater than the infinity of natural numbers? What is the rationale behind this absurd distinction?
Infinity and Beyond
According to the Banach-Tarski paradox, it is possible to divide a sphere into five parts and reassemble them in a peculiar way to produce two copies or replicas of the same sphere. Bear in mind that the parts are reassembled, not stretched, and yet they produce two spheres of the same area, volume and density. The paradox entails the most daunting of all mathematical atrocities: 1=2.
However, the paradox cannot be realized in the real world, but only in the strange and phantasmagorical realm of infinities. The paradox assumes that the sphere is not composed of rigid atoms, but points, infinite points, each of which is exempt from the laws of physics, but not mathematics. And the laws of mathematics can be profoundly strange – strange enough to allow the sum of all positive natural numbers, the sum 0+1+2..∞ to be equal to -1/12.
Buzz Lightyear of Toy Story is embarrassingly incorrect when he affectionately declares “To infinity… and beyond!”. Infinity has no end that one can surpass; the end, in fact, doesn’t exist. If you count to x and believe that you have reached infinity, add one to x and you will witness the emergence of a new, greater infinity. Then add one to this new infinity and witness the emergence of an even greater infinity. Divide this infinity by two and you are still left with infinity. This is the very principle on which the Banach-Tarski paradox is based.
Proverbially, infinity is considered to be just like any other number, a quantity denoted by an ‘8’ sleeping on its supple back. However, this is not true; the x only exists in the abstract, as a mere idea. As explained above, you can only – provided you have the vigor and time to count – approach infinity, but never reach it. It is infinity’s paradoxical nature that has frustrated us since antiquity; it is the mathematician’s oldest nemesis.
However, we have only dealt with the countable infinity. The infinity of natural numbers is finite because it is at least fathomable, but prepare to tussle with the unfathomable: uncountable infinity – the infinity of real numbers.
Write on an infinitely large page an infinitely long trail of even numbers. Replicate this trail, but now write every odd number in between the even numbers. A mathematician would claim that, believe it or not, the two trails are equally long, since infinity divided by two is still infinity.
However, the space between 0 and 1 can be divided into one hundred parts: 0.01-0.99. Furthermore, the space between 0 and 0.01, can be divided into another hundred parts: 0.0001-0.0099. In fact, one can do this, you guessed it, an infinite number of times.
The notion is so mind-bogglingly ridiculous that readers should realize by now that there exist more real numbers between 0 and 1 itself than there exist natural numbers. By performing the division between every natural number, one would discover that there exist infinite numbers between every natural number, which themselves are infinite. The resulting trail would be infinite times infinite long. The infinity of real numbers is unfathomably large, to the extent that mathematicians call it the ‘uncountable infinity’.
This is not mathematical hocus-pocus; it is logically consistent. The German mathematician Georg Cantor was the first to verify the uncountability of real numbers with a rigorous proof. What Cantor basically did was draw two columns, one for natural numbers and the other for real numbers. The natural number in each row was assigned or matched to a unique real number. Obviously, if the rows are found to be equal in number, the two infinities are equal.
Cantor then performed what is now called diagonalization. He incremented the real numbers by one, but at particular places, according to a rigorous algorithm. If you increment the real number column diagonally in a certain quirky way, completely new and unique real numbers are produced. The two rows, he proved, are not equal. The infinity of real numbers is undoubtedly greater than that of natural numbers.
What fascinates me is the conundrum of whether nature really permits this, whether these infinities actually exist or have we, helpless slaves to logic, taken it to its bitter end? Do logic and numbers really exist in nature or do they exist only in the abstract, only in our minds?