geometry template for infinite-line-ray to precision define infinite-number in Peano Axioms #627 Correcting Math
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From: Archimedes Plutonium on 3 Jul 2010 15:39 Alright, making some progress here even though some backpeddling. Geometry has precision definitions of finite-line versus infinite-line and they have two types of infinite-line, the line-ray with one arrow and the line with two arrows. But Geometry does not imply or hint or suggest that there must be a boundary between finite line and infinite line, unless we realize that you cannot build a infinite-line ray without using infinite-number. And you cannot define infinite-number unless you select from physics a number where there is no longer any Physics meaning in the Universe. That involves the Planck Units and 10^500 has no more physics measuring allowed. So does Geometry offer a template for Peano axioms and Algebra to define infinite-number? Yes it does. For if you examine the Peano Axioms and insert an additional new axiom of this: Axiom: numbers are finite if less than 10^500 and any Natural Number of 10^500 and larger is an infinite-number. So now, let us look at how Geometry lines are template, that means they are similar in meaning and in building, are the template of Numbers in Algebra. The two numbers 0 and 1 as endpoints, in Peano axioms would be a line segment 0 to 1. The two numbers 1 and 10^500 as endpoints forms a line- segment also, a finite-line, since it is less than 10^500. However the two numbers as endpoints 0 and 10^500 forms not a line-segment but an infinite-line-ray since it has a length equal to or greater than 10^500. Now the two numbers of 5 and ((10^500) +3) forms a line segment because the length is less than 10^500. So we see here that Geometry serves as a template of how to precision define finite-number versus infinite-number. Now in the organic building of the Peano Natural Numbers to the Rationals then to the Reals, in that building process we included the negative numbers and with their inclusion we use the Geometry template to correspond with the infinite-line with its two arrows. So that an infinite-line corresponds with (-)10^500 to that of (+)10^500. I am not familar with the actual history of mathematics and geometry in particular and this information is probably lost in ancient times. The information as to whether we first discovered, and obviously true the finite-line as a line segment. And then the next line discovered would have been, according to this post, the next line discovered would have been a infinite-line-ray with its single arrow in a direction. The discovery of a infinite-line with two arrows would have been the last discovery of lines in geometry. History probably lost that information, but that is what makes common sense. To think that the progression of discovery of different lines in geometry went from finite line to that of infinite-line of two arrows suggests that we would have known 0 exists as a number and that the negative numbers existed in the very early development of mathematics, yet we know as a fact that 0 took a long time to be recognized and understood and even longer for the negative numbers. So that not until we had infinite-line-ray long time established would we ever have the double arrowed infinite-line discovered. This would have been centuries before the Euclid parallel postulate where a infinite-line double arrowed was required. Even then, the infinite-line-ray would have been sufficient. So far, I have only been able to ascertain that a boundary between finite and infinite is necessary in Algebra and Number theory in order to build a infinite- line in geometry from that of finite-lines. So far I needed both Algebra and Geometry together to assert that a boundary must exist. But can I find other areas of mathematics that require this boundary between finite and infinite? Another example that could also demand that Algebra and Geometry must have this boundary between finite and infinite. It is the old paradox of the turtle and rabbit race, called Zeno's paradox, that the turtle is given a head start lead and according to the paradox the turtle wins the race. The explanation for it is of course Physics that the rabbit wins due to the concept of speed. But when mathematicians try explaining this paradox with the infinities of small distances, it is never a satisfactory explanation. But now, let us inject into that explanation in the turtle rabbit race there are only four distances of 1, 2, 3 and 4, where the turtle starts the race on 2 and only has to reach 4, and where the rabbit is on 1 and has to reach 4 to win. So that by the time that the turtle reaches 3, the rabbit is already on 4. Here we begin to see that if mathematics has no boundaries between finite and infinite such as the boundary of 10^-500 where there are nothing but holes and gaps between numbers this small or smaller, then we begin to realize that Zeno's paradox is a reflection of the fact that mathematics has no absolute continuity. What I am trying to say is that Geometry really does have a boundary between finite and infinite otherwise the Zeno rabbit and turtle race would not be a paradox. Here I am talking of micro infinity 10^-500, not the macro infinity of 10^500. So is not the Zeno paradox really about the idea that there must exist a boundary between finite and infinite. Summary: in the macro world, we must have a boundary between finite and infinite in order for geometry to be able to construct a infinite line from finite lines, otherwise no amount of finite lines could ever be a infinite line. In the micro world there must also be a boundary between finite and infinite otherwise the turtle truly wins all the Zeno races. 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 |