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 Infinite tetration and superroot of infinitesimal andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 01/04/2008, 06:51 PM jaydfox Wrote:infinitesimals would necessarily have density measured with 2^CIf this is true, then there cannot be an isomorphism between the standard field of complex numbers $a+bi$ where i is imaginary unit (Cardinality of the reals), and the non-standard field of $a+bz$ where z is an infinitesimal (Cardinality of the powerset of reals), as Ivars suggests. If they are of fundamentally different cardinalities, then an isomorphism between the two is impossible. @Ivars: This would mean you have to make the distinction between the imaginary unit and an infinitesimal. jaydfox Wrote:I never bought the distinction in density ... between rationals and realsIn my mind, there isn't any distinction in density, only in countability. Intuitively, it makes sense that the rationals are countable and the reals are not, but all other distinctions like power-sets and subsets and cardinalities and stuff, just confuses me. Andrew Robbins andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 01/04/2008, 07:08 PM Never mind. I looked it up. $|*\mathbb{R}| = |\mathbb{R}| = c$. Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 01/04/2008, 09:48 PM andydude Wrote:jaydfox Wrote:infinitesimals would necessarily have density measured with 2^CIf this is true, then there cannot be an isomorphism between the standard field of complex numbers $a+bi$ where i is imaginary unit (Cardinality of the reals), and the non-standard field of $a+bz$ where z is an infinitesimal (Cardinality of the powerset of reals), as Ivars suggests. If they are of fundamentally different cardinalities, then an isomorphism between the two is impossible. @Ivars: This would mean you have to make the distinction between the imaginary unit and an infinitesimal. Andrew Robbins I do... But that will confuse things further: Imaginary unit is a universal notation for infinitesimals if we do not care about differences they have in different scales. In each scale though, infinitesimals are different... So they have more properties than just imaginary unit, internal degrees of freedom which i hope will be revealed by tetration. A scale: A scale in my interpretation is what you get when You differentiate/integrate function using infinitesimals instead of limits: dy and dx is infinitely smaller than x and y. If You differentiate y(x) the resulting function y'(x') (x' to remember scale has changed) is in fact, in infinitely smaller scale than y(x). But again, if You can differentiate it further , there is dy'/dx' which are again infinitesimals in the new scale, so the resulting y''(x'') lays in 2 infinities smaller scale than y(x). In scale where y(x) was, or even y'(x'), y''(x'') is nothing. 0. But it is not the smallest nothing in the world, since there are scales infinity below that. The same goes for integration, in opposite way. Differentiation usually includes information loss about absolute placement of function in space. Integration recovers the scale, but information about placement is arbitrary-chaotic in a sense- it kind of wanders all over place - hence infinite speed since as we provide information about initial conditions, integration places function immedeately where it should be. Now, if we ignore the fact there are infinitely different scales- and in mathematics we do, we may operate with imaginary unit over all scales as if there were no difference. e takes care of that. But actually, in each scale infinitesimals are different. An infinitesimal if looked from the scale we are in y(x) , is imaginary, since it does not exist in that scale, as it is 0, but it does exist in next or other scales , where it is finite. So maybe You are right- normal i is an averaged infinitesimal over all scales ( as long as it is defined (or complies with) e^ipi/2+-i2pik. ) But each of values of i as it rotates by infinitesimal angles from one scale to next changes in complexity and matches pi and 1/2 ( I am not sure about e) on that scale and angle, providing a finite angle. So e.g i= (i-1)/(i+1) = h(e^pi/2) defines only one infinitesimal in one scale depending on e, pi, 1/2. We can not distinguish them as for todays math, all e, i,pi, 1/2 are the same, but they are different in each scale, just results in the same angle. See how bad it gets.... Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 01/05/2008, 09:38 AM Hi Andrew To expand slightly: Numbers are being built as part of informational world organizing itself. Not all numbers exist in smallest scales, and , as they are built, they acquire the properties they have by including the process of their creation- so Numbers as such are much more complex entities than we think. They have a history, and a scale where they naturally appear. On more thing is non-linearity - may still early- but in my view, pixels which has complex internal infinitesimal structures, can have at least 2 additional properties: 1) depth- how many scales below are fixed to that point - it can be 1, it can be many- so pixel can be fixed in its place just in 1 scale, or many. If entity is described by differential equation, that would be the order of differential equation. 2) Non-linearity - entities can be stretched and compressed creating variations that make each place almost unique in space, increasing or decreasing their Reynolds number ( ratio between content of information inside and exchange with flow around them) and that changing their inertia and speed of information transfer- including speed of light and change of gravity field in scale just below quantum scale. Again, if entity represents a differential equation that works on infinitesimals, this means it is non-linear. So every process or structure of infinitesimals described by differential ( and integral) equation involves as many scales simultaneously as is the order of differential equation. I see differential equation as a machine ( differential combination) that connects more scales in a given place in space then just 2. Linear differential equation processes only 2 scales of infinitesimals, so that is the complexity it can represent. And we do not know, usually, between which 2 scales we look at the problem , in mathamatics. In physics, we usually know. 27th order continuous differential equation is a structure of different complexity differentation machines at different infinitely distant infinitesimal scales below quantum scale which simultaneously connects 28 scales, it works on them , kind of lets them flow through it while itself not changing or perhaps, changing itself- adaptive differential equation. So we can see that first: we do not understand numbers as we do not know how they form as complexity grows Second- we do not understand that differential equation are relative , and first order differential equation will always connect only 2 scales of different complexity, but which 2? Non-linear differential equation of 4 order might already fit something interesting because i has a cycle of 4 , and 168th order non-linear differential equation applied to 168 SCALES below quantum might have very interesting and exact properties. If we include above quantum world, we have to be careful to keep the infinitesimal scales involved pertinent to addressable problem, plus mix discrete difference and continuous differential equations to describe the system we are looking at, checking against Reynolds numbers at each scale if we want to make approximations. So now differential/integral equations has been added to the picture as well andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 01/05/2008, 07:37 PM Ivars Wrote:Imaginary unit is a universal notation for infinitesimals if we do not care about differences they have in different scales. No no no! The imaginary unit is $\sqrt{-1}$ whereas i is just a letter. Do not confuse the two, one is a letter, one is a number. Feel free to use the letter i for infinitesimals (or for whatever you want), but when you do so, do not call it "imaginary unit" call it "the letter i". Andrew Robbins andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 01/05/2008, 07:49 PM Also, if it is "universal notation" as you say, then can you provide some references? So far all references I've encountered about infinitesimals and hyper-reals use $\delta, \epsilon$. Andrew Robbins Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 01/05/2008, 10:30 PM andydude Wrote:Also, if it is "universal notation" as you say, then can you provide some references? So far all references I've encountered about infinitesimals and hyper-reals use $\delta, \epsilon$. Andrew Robbins Yes it is imaginary unit =sqrt(-1) which is infinitesimal, in my thinking. Imaginary infinitesimal angles. That is a speculation. Infinitesimals of length are dx, dy. No, I can not provide . The closest I have seen is this link from prof. Bell, and it involves epsilon - so far I have not been able to find a place for it in a scaled math. i*epsilon is infinitesimal imaginary. In my opinion, infinitesimal angle with infinitesimal length. But I am not sure. http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf Quote:But spacetime theory in SIA forces one to use imaginary units, since,infinitesimally, one can’t “square oneself out of trouble”. This being the case, it would seem that, infinitesimally, Wheeler et al.’s dictum needs to be replaced by Vale “ic(t)”, ave “iε” ! andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 01/06/2008, 01:10 AM Quote:Yes it is imaginary unit =sqrt(-1) which is infinitesimal No, that's nonsense. $(\sqrt{-1})^2 = -1$ where as $(dx)^2 \approx 0$ so they cannot be the same thing. If they were of slightly different "scales" then I would be more inclined to agree, but 0 and -1 are nowhere near the same number! This is a contradiction that proves your assumption $\sqrt{-1} = dx$ is false. Andrew Robbins andydude Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 01/06/2008, 09:42 AM Oops, I just realized that $(p \wedge \neg p)$ is one of those things that just does not hold for infinitesimals. How convenient. Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 01/06/2008, 04:54 PM andydude Wrote:Oops, I just realized that $(p \wedge \neg p)$ is one of those things that just does not hold for infinitesimals. How convenient. Which means...? « Next Oldest | Next Newest »

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