At 12:33 AM 10/7/2005, you wrote:
A measure is a function m(x) on your set obeying additivity:
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,
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.