Infinity

Infinity, it was thought, simply referred to a phase of numbers at which counting could not be done. We now know that it is a lot more than simply that. Infinity is a mathematical concept with its own set of rules and it's an undefined area of arithmetic; a bit like dark matter in physics. We know it exists but it eludes our area of understanding. For example, the sum of numbers as follows: 1+2+3+4+5......= -1/12. This sum of course, makes no sense as the sum of numbers up to infinity should obviously be infinity, but derivations have shown the above result to be true.  Another famous concept is Hilbert's Hotel Paradox, where a hotel has an infinite number of rooms with as many guests. As an infinitely larger number of guests enter the hotel, each guest must move up a room to accommodate more guests. However, Georg Cantor said that the power set of a set has more elements than the main set itself. This was known as Cantor's theorem. This is applicable in the paradox as it proves that there is in fact something greater than infinity, called the continuum, and it states that if a hotel has infinite rooms, there cannot be space for more than infinite guests even though it may seem paradoxical. Another application of the eeriness of infinity is Zeno's dichotomy. It states that in order to get from point A to point B, one must first reach the half way point, and before that the 1/4th point, and before that the 1/8th point and so on; basically one has to accomplish an infinite number of tasks before completing the main task. However, since we know that the sum of an infinite number of terms has a finite result, this paradox was disproven. In physics as well, the chaotic boundary conditions theorem states that there can either be an infinite number of universes, or our universe is finite but without a boundary. In chess, it was postulated that with each move of a game, the number of games increased exponentially and drastically to a very very large finite but seemingly endless amount as each move represented a new game, a new universe, per se. Other concepts of infinity are Thompson's lamp( a variation of Zeno's paradox), and Galileo's paradox, which stated that the list of all natural numbers would be greater than the list of squares as an infinite number of numbers was greater than an infinite number of squares, but since each number has a square, and that number is the square root of its square, numbers and their squares are equal in number. The postulates are endless(pun intended).