Hi Russell: At 12:33 AM 10/7/2005, you wrote:

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A measure is a function m(x) on your set obeying additivity: m(\empty)=0 m(A u B) = m(A) + m(B) - m(A^B) where u and ^ are the usual union and intersection operations. The range of m(x) is also often taken to be a positive real number. Does this answer your question? Measure is generally speaking unrelated to cardinality, which is what you're referring to with finite, countable and uncountable sets.

`I have looked at this and several other sources of a definition of`

`measure and some example computations of measure. As a result I do`

`not at this time see that there is a function on my set that can be`

`additive since my set consists of fragmentary`

`descriptions. However, I am still interested in the relative`

`frequency of finite length vs countably infinite length descriptions`

`- as assembled from the fragments by dividing the list into to sub`

`lists - that are in my All. Then working with this to produce some`

`idea of the relative frequency of worlds that can appear to an`

`observer to be well behaved. To start I have edited [corrected?] my`

`original post below and then go from there.`

On Wed, Oct 05, 2005 at 10:45:28AM -0400, Hal Ruhl wrote: > > In my model the ensemble of descriptions [kernels in my All] gets > populated by divisions of my list of fragments of descriptions into > two sub lists via the process of definition. > > The list is assumed to be countably infinite. > > The cardinality of the resulting descriptions is c [a power set of a > countably infinite set] > > Small descriptions describe simple worlds and large ones describe > complex worlds. >

`For each natural number n there should be countably infinite [is, is`

`not] pairs of descriptions of lengths [n, countably infinite]. There`

`are countably infinite n's. There are also countably infinite [is,`

`is not] pairs of descriptions of lengths [countably infinite,`

`countably infinite].`

`Again I am not a mathematician but if the above is correct I think it`

`means that there are as many finite length descriptions in my All as`

`there are descriptions that are countably infinite in length.`

`Many of these descriptions regardless of length are just random`

`collections of items on the list and not likely to describe states of`

`universes that can sequence so as to produce evolving universes a SAS`

`might preference [select?]. I suspect that all of the countably`

`infinite length descriptions are not SAS friendly.`

`However, there are just as many finite descriptions of states of`

`universes that can sequence in SAS comfortable ways [even with the`

`True Noise in my model].`

`My conclusion so far is that finding ourselves in a universe like the`

`one we do is not the least bit out of the ordinary.`

Hal Ruhl