using geometry's finite-line & infinite-line and line rays to hopefully well-define finite-number & infinite-number #623 Correcting Math
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From: Archimedes Plutonium on 2 Jul 2010 13:44 So let me try going from geometry's precision well-defining of finite- line versus infinite-line to see if I can well-define Algebra's numbers. I do not expect to make progress on this because geometry has three different types of geometry with its NonEuclidean elliptic and hyperbolic. One thing is sure, that if we posted a list of unsolved problems, over 90% come from Number theory Algebra and less than 10% come from Geometry. This is witness to the fact that the concepts over in geometry are pretty much well defined, especially finite line and infinite line. For starters, I defined AP-adics as for example 9999....9997 with frontview and backview and infinity in the middle. And Hensel P-adics has .....999999 = -1. Now in Euclidean plane geometry we have a line-segment, or finite-line such as __________ and we can consider it as say 0 to 9 where we have 0_________9 where the 0 and 9 are endpoints, or we can say it is from -3 to 6 such as -3___________6 And in geometry we have a infinite line-ray as ---------> where the endpoint could be any number, finite-number since in old-math that was the only type of number given in Peano Axioms. And the arrow end of that line-ray is what? Well the arrow end cannot be the frontview nor the backview in 9999....9997 of AP-adics, but the arrow can be the dots of the Hensel p-adic ...... 99999 but that is sort of messy since that number is -1 afterall. So none of this really is working out, is it. So now let me inject the definition that 10^500 is the boundary between finite number and infinite number. Now given the infinite line ray of -------- >, does it make any better sense? Perhaps so, if we stipulate that the endpoint is 0 and that the arrow is beyond 10^500. Or, the endpoint is 10^-500 and not zero. And then what is a infinite line such as <---------->, is it from (-)10^500 arrow to (+)10^500 arrow? So there is alot of complications and mischief here. The translation of infinite line ray and infinite line to a precision definition of infinite-number for Algebra is not working smoothly, is it. There are some bumps in the road of this adventure. So in cases like this, we should defer to the king of sciences-- physics. What would physics say is a precision well defined infinite number? Well physics needs negative numbers, especially for electricity and magnetism, so then the infinite number should have an arrow if it exceeds either the positive 10^500 or the negative 10^500. So in this fashion, the Hensel p-adics do not qualify as infinite-numbers because ....99999 is -1 and we easily measure or count to -1. The AP-adics fair even worse than the Hensel p-adics. Again, what the 10^500 boundary does, is show us that it is the only way for Number theory to define finite number versus infinite-number and still tie back or connect with geometry's definition of finite line versus infinite line. Any other system of infinite number such as AP-adics or Hensel p-adics are artificial and flawed. For years now I have been harping that the Hensel p-adics are merely a narrowed disguised set of restricted Reals, or a subset of Reals with a fancy operation. That the Hensel p-adics were not an independent set of new numbers. But the AP-adics, although independent new set of numbers, are they realistic or fiction? Apparently they are fiction and have no reality. The number 10^600 is an infinite number when 10^500 is the boundary and both numbers have place values, but the AP-adic number 99999.....99997 or 333....4444 has no place value, and never will since 10^500 means infinity and so the dots in 99999.....99997 mean 10^500 dots, thus absurdity. So Hensel p-adics cannot be infinite numbers, nor can AP-adics. Both those systems are either fictional sets or as in the case of Hensel p-adics, a disguised subset of the Reals. When we have the boundary between finite and infinite as a number exceeding 10^500, we have to throw out or discard both the Hensel P-adics and the AP-adics. Both those number systems were devised with a misunderstanding of infinity. Infinity means going beyond a certain point. Going beyond where Physics can no longer measure or count or do anything. If physics is not in the picture, then it is infinite. So the arrows in an infinite line ray or an infinite line mean that there is no longer any physics going on, or we cannot reach that physics. Infinity means more of "unknown physics" rather than "endless". The concept of "endless" is like the concept of the "present bald king of France" or the "fire breathing dragon". Endlessness and continuity were never realities in physics and so are fictions. Enough said for one post. And note, in this post I rejected mine own AP-adics, which I rejected a long time ago when I went with the 10^500. But I never publically declared the rejection of AP-adics. Here I declare I have thrown them out as fiction. And it goes to show we keep things or ideas, only when there is nothing better to replace them. And it looks as though the only infinite-numbers definition that makes sense is to call numbers a little larger than 10^500 such as 10^600, but that any other system such as 6666....88888 or .......11111 is just folly. 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 |