```Date: Nov 30, 2012 11:42 PM
Author: ross.finlayson@gmail.com
Subject: Re: Cantor's first proof in DETAILS

On Nov 30, 11:42 am, FredJeffries <fredjeffr...@gmail.com> wrote:> On Nov 30, 8:39 am, "Ross A. Finlayson" <ross.finlay...@gmail.com>> wrote:>>>> > > You've had this function for 13 years now and you STILL can't> > > calculate the area of a triangle with it.>> > Fred Jeffries who I respect:  I'd like to think that's in the context> > of modeling Dirac's delta with triangles or radial basis functions,> > but what's important to describe of EF as plotted is this:  removing> > all the space between the integers and plotting the elements in the> > range it would look like f(x) = x from zero to one, half a square and> > a triangle, but the F-Sigma Lebesgue integral of EF evaluates to one> > not one half, now that's the surprise.>> > EF:  CDF:  of the uniform distribution of the natural integers.>> Sorry, I can't decipher the above two paragraphs. All I see is> 13 years and 3 math degrees and still can't calculate the> area of a triangleThe area of a triangle is base times height over two.A CDF ranges from zero to one over the range of the elements in thestatistical/probabilistic distribution and is increasing.  A uniformdiscrete distribution would have for any m, n that CDF(m+1) - CDF(m) =CDF(n+1) - CDF(n), constant monotone.  Where EF is this CDF,putatively, 1/d = 1/d which is true, satisfying these requirements.The notion of a uniform probability distribution over all the naturalsis not necessarily intuitive, and I described how to build one in ZFCbesides that EF has the concomitant properties of being the CDF of auniform distribution of the naturals.The reference to real functions modeling Dirac's delta a.k.a. the unitimpulse function is that this function is a spike to infinity at zero,zero elsewhere as defined in the reals, whose integral evaluates toone.  It's standardly modeled as triangles or radial basis functionsor any other function really that have area equal to one anddiminishes to point width at zero as parameterized by an unboundedfree variable.  Similarly Heaviside's step is so modeled with aparameterized arctan() and etcetera.Here, EF's family of functions so modeling it is simply parameterizedby d as it is unbounded.Then, I went deeper to the foundations than that.  Simply working up amutual definition of the real numbers as constructively at oncecomplete ordered field, and, partially ordered ring, with, ratherrestricted transfer principle, as for example we know from Cauchy/Weierstrass and Bishop/Cheng, then, it's possible to have thecomprehension of the function as a: primitive function, in fact_defining_ the unit line segment.  A corresponding geometry of pointsand spaces to complement Euclid's of points and lines is initiallydefined, with a fundamental space-filling curve defining shapes viasimple properties.Then of course there are the set-theoretic results extra the number-theoretic results re: cardinality, an axiomless system of naturaldeduction with natural definitions of sets and ordinals followingdeductively gives a theory with an empty and universal set in thedually-self-infraconsistent dialetheic and paraconsistent.So, yeah, in the time between noting the simple construction of EF andtoday, there's quite a bit of development.  Dogged determination, asit were, for me partially satisfied in a great appreciation of thefundamental philosophical tenets.No, I only have a Bachelor's of Science degree (in mathematics thankyou and I know computer science).  The guy who wrote a dissertation toconvince soi-disant set theorists that half the integers are even hasa Ph.D. from M.I.T.  He got it for writing a dissertation in settheory that half of the integers are even.I wonder your familiarity with Nyquist, Shannon, Huffman et alia andhow Nyquist's sampling theorems in the discrete would apply, here tothe continuous or to sets dense in the real numbers.  I don't know ofmuch work in that area.And Fred Jeffries, I respect you even where you claim not to makesense of this, thank you please for not making no sense of it.Basically these notions are very fundamental to what is continuous andwhat is discrete.So, matter as the atom is particle and wave.  What then is our simplepoint?Regards,Ross Finlayson
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