```Date: Mar 18, 2013 3:32 AM
Author: fom
Subject: Re: Matheology § 224

On 3/18/2013 2:10 AM, Virgil wrote:> In article <mOOdnU1k7dJ5ONvMnZ2dnUVZ_rednZ2d@giganews.com>,>   fom <fomJUNK@nyms.net> wrote:>>> On 3/17/2013 7:11 PM, Ross A. Finlayson wrote:>>   >>>> A simple and trivial>>> continuous mapping was noted.>>>>>> Regards,>>>>>> Ross Finlayson>>>>>>>>> That is not enough Ross.>>>> By definition, a linear map must satisfy>>>> f(x+y) = f(x) + f(y)>> f(ax) = a*f(x)>>>> So, the domain must at least have the>> structure of a module since it needs>> to have an abelian addition of domain>> elements and a map from the domain>> into itself with a scalar multiplication.>>>> Furthermore, it is unlikely that one>> could take the scalar multiplication>> to be the Galois field over two>> elements since multiplication by>> zero would be the zero vector and>> multiplication by one would be>> the identity map.>>>> A morphism with that scalar field>> could not reasonably be expected>> to have a linear map with a>> system of real numbers.>>>> In order to build a scalar that>> could even possibly serve this>> purpose, given WM's claims related>> to various finite processes, one>> would have to invoke compactness>> arguments involving completed>> infinities.>>>> For example, for any non-zero>> sequence of zeroes and ones>> that becomes eventually constant>> with a trailing sequence of zeroes,>>>> 1001101000......>>>> we can replace that sequence with>> a trailing sequence of ones,>>>> 1001101111......>>>> We want to use these forms because>> of the products>>>> 1*1=1>> 1*0=0>> 0*1=0>> 0*0=0>>>> Then, coordinatewise multiplications>> along the trailing sequence of ones>> retains a trailing sequence of ones.>>>> In addition, on the interval>>>> 0<x<=1>>>> we can associate 1 with the constant>> sequence,>>>> 111...>>>> Given these facts, we can now say that>> a collection of infinite sequences is>> "compactly admissible" if for every>> finite collection of those sequences>> coordinatewise multiplication yields>> a sequence different from one>> consisting solely of an initial>> segment of zeroes followed by>> an initial segment of ones.>>>> In other words, even though>>>> 000000111...>>>> may be representationally>> equivalent to>>>> 000001000...>>>> for some purposes, compact>> admissibility has to ignore>> what happens in this conversion.>> The situation above is>> interpreted as corresponding>> with a non-compact set of>> sequences.>>>> Given this, sequences like>>>> 1000...>> 11000...>> 110000...>> 1101000...>>>> yield>>>> 1111..>> 11111...>> 110111...>> 1101111...>>>> whose coordinatewise product>> is>>>> 1101111...>>>> So that the original sequence>> is compactly admissible.>>>> Given a construction along these>> lines, one could then think of>> compactly admissible collections>> as possibly forming a sequence space>> as described here>>>> http://en.wikipedia.org/wiki/Hilbert_space#Second_example:_sequence_spaces>>>> Obviously, the compactly admissible>> collections are not defined as>> converging in the sense of a sequence>> of partial sums.>>>> Equally obviously, I have not done>> all the work necessary to decide>> whether or not this would work.>>>> My purpose here is to explain that>> the scalar multiplication would>> require a construction along these>> lines just to even begin to talk>> about whether or not WM could>> do what Virgil is asking.>> MY points are>> (1) The bijective mapping from the set of binary sequences to the set of> paths of a Complete Infinite Binary Tree, was NOT a linear mapping as it> was originally formulated by WM.>> (2) WM is  not competent enough to be able to reformat it correctly .> i.e., to make it an actual and obvious linear mapping.>> (3) It is not all that difficult to create an actual and obvious linear> mapping there for someone who knows something more about linear spaces> that WM does.>Since you use the word "obvious" I am assumingthat I did much too much work attempting toestablish a scalar multiplication.But, I would assume that you would bebasing it on the representations as realnumbers directly. I tried to avoid thatbecause of WM's finitist claims and theabstract definition of tree paths.
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