Thursday, 15 November 2012

Infinity- The Sky's Not the Limit

The sky's the limit. Such is a cliche that dares to be challenged. From the very breadths of the natural numbers to the measure of human stupidity, the notion of infinity seems simultaneously intoxicating and hair-raising. With paradoxes, anomalies and incongruities lurking round every avenue and boulevard of its grandeur, it remains feared and revered, not as a stream of digits but as a reality in itself. Infinity denotes the antithesis of the finite, the extension of the boundary and the never ending twist to the tale; a metaphorical and mathematical playground most thinking people discover for themselves. Just imagine a hotel with an infinite number of rooms and an infinite number of guests, if an additional guest arrives, the existing guests must shift into the next room to accomodate him. Which represents one of the great paradoxes of the world of infinity, infinite sets are in fact proper subsets of themselves. But what if a second set of infinite guests arrived? The existing guests must shift into the room with double the value of theirs. And what if all of the guests but one left in the evening? It seems that infinity minus infinity can be anything you like. And when comparing two infinite sets, just like in the hotel allegory, they are both the same size even though one set appears to contain half as many numbers. Indeed this represents quite a mouthful for a reality that stretches minds beyond imaginable reach. Thinking diagonally about infinity, we arrive at an intriguing impasse; the infinity of the decimal numbers is bigger than that of the counting numbers. But beyond the puzzles and counter-intuitives, the realm of infinity seems under fire by the theory of a finite number scheme; one where the eccentricities of  infinity are dispelled by an ultra-finitism of a biggest number. Indeed, if there was a biggest number, the 'add one' doctrine would land you right where you begun, at zero. Such a seemingly natural, elegant and lucid idea is purely questionable, quite comparable to dividing by zero in elementary algebra. But regardless of the whether there are finite of infinite sets of googols, googlepexes or even googolplexians; there seems a natural necessity for an infinity. Despite the absence of a physical, tangible or natural expression of infinity in the grand scheme of things, such is a conceptual metaphorical and mathematical truth, rather than an illusionary mirage of the mind. Born via an innate human impulse to go above and beyond any frontier and bred by the loops and twirls of its bewilderment and perplexity. One, two skip a few; to infinity and beyond

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