infinity on the small scale between two integers and infinity on the large scale #628 Correcting Math
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Next: appending the new axiom to the Peano Axioms that precision defines finite versus infinite #629 Correcting Math
From: Archimedes Plutonium on 4 Jul 2010 02:41 Funny how infinity on the large scale, such as infinity of counting numbers, yet with defining infinity as starting with 10^500 really does not stir up the mathematicians like a hornet's nest. Sure a few will come out of their hive or clam shell and take a peek at what is going on, but then right back to the ivory tower sipping more coffee. But then when they hear that 10^500 as the boundary between finite and infinite causes there to be no more absolute continuity of Reals less than 10^-500, well, that is a hornet's nest of mathematicians buzzing around all over the place. Funny how talk of large infinities is easily ignored as idle talk, but once you talk about small infinities, then you are treading over limit concept and convergence and even the definition of Reals as Dedekind cuts. Now the proof I gave a few days ago that you must have the boundary between finite and infinite-number and 10^500 is a reasonable choice, the proof that you cannot build a infinite line in geometry from any amount of finite lines unless there was a boundary to denote infinite-numbers. So you are stuck in geometry when Algebra never defines infinite-number versus finite-number. Stuck in not being able to construct an infinite-line. But much earlier, I forgotten if 2009, I showed where absolute continuity in the small scale of between two consecutive integers, that if absolute continuity existed, then one can construct a triangle that has two angles of 90 degrees, and thus more than 180 degrees. Another supporting evidence that you cannot have absolute continuity comes from Quantum Mechanics of physics were energy, distance and time are quantized, meaning whole multiples with emptiness in between. Quantum Mechanics means gaps or holes in between. So the pursuit of absolute continuity in mathematics for centuries or milleniums, or for how long exactly has mathematics been concerned over continuity, seems rather funny and misplaced effort, because Quantum Mechanics is all about discreteness and thus holes and gas in between. So it looks as though I have now two proofs that a boundary exists between finite versus infinite lines and a boundary exists between finite versus infinite numbers. Proof One is that you cannot build a infinite-line-ray without there being a boundary in numbers where the next number and all larger are infinite numbers, such as 10^500. Proof Two is that if you have no boundary between finite versus infinite lines, or finite versus infinite numbers in the small scale of mathematics, then you can construct a Euclidean triangle which has two right-angles and the sum of angles is greater than 180 degrees. Proof Two thus implies that if you have a boundary between finite and infinite, whether lines or numbers, implies that you cannot have absolute continuity in mathematics. So, there is some really heavy mathematics here with this issue of precision defining finite versus infinite. Archimedes Plutonium http://www.iw.net/~a_plutonium/ whole entire Universe is just one big atom where dots of the electron-dot-cloud are galaxies |