Re: A question re measure {correction 2}
Hi Russell: I would change my last post and say that since tails pair with many heads and heads pair with many tails [assuming I am right re what you said] the most compact way to build an All is to use heads as the kernel where ever possible. As heads are encompassed by evolving Somethings the head and all its tails are given the instantation of reality. While the many tails will mostly describe chaotic universe states there is always an associated head. I think this makes the picture one where - given enough simultaneous Somethings - there is always a head that is able to provide a consistent extension of any universe state sequence and simultaneously having reality. There should always be an infinite number of concurrent Somethings. So again I think a well behaving universe should not be out of the ordinary. Yours Hal Ruhl
Re: A question re measure {correction}
Hi Russell: Correction in caps and []. More Additional comments: My definition of kernel is the information necessary to establish a particular division of the list. This requires the All to contain mostly tails if my comments in my last two posts is ok. The result of my dynamic as I see it is that specific finite heads are given moments of physical reality far more often then I originally thought. This allows more consistent histories of sequences of prior states to have consistent extensions [IN ANY GIVEN STEP] of the dynamic. Thus evolving universes with finite descriptions of their states may actually predominate. Yours Hal Ruhl
Re: A question re measure
Hi Russell: More Additional comments: My definition of kernel is the information necessary to establish a particular division of the list. This requires the All to contain mostly tails if my comments in my last two posts is ok. The result of my dynamic as I see it is that specific finite heads are given moments of physical reality far more often then I originally thought. This allows more consistent histories of sequences of prior states to have consistent extensions. Thus evolving universes with finite descriptions of their states may actually predominate. Yours Hal Ruhl
Re: A question re measure
Hi Russell: Additional comments: I think what you mean is that each specific head is paired with many tails. Since only one side of the pair is required in the kernel ensemble my model would need to be modified to the effect that both sides of the pair is given physical reality whenever a kernel is given physical reality. I find this satisfying from symmetry and informational points of view. Yours Hal Ruhl
Re: A question re measure
Hi Russell: At 07:48 PM 10/8/2005, you wrote: On Sat, Oct 08, 2005 at 12:26:45PM -0400, Hal Ruhl wrote: > > 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]. > I don't think this is right, but I could be grasping the wrong end of the stick. I think of your definition division as the division of an infinite length symbol string into a finite head, and a countably infinite long tail. Ok If true, then there are A^n heads of length n, and c (=A^\aleph_0) tails. Therefore, there are c pairs of descriptions. Ok, if you mean that there are c pairs of descriptions in which one of the pairs is of length n etc. etc. I find this [I think] even more satisfying than my above. However, I see the basic result as being the same i.e. the number of descriptions of any particular type is always c so there is no preponderance of any type of description. Yours Hal Ruhl
Re: A question re measure
On Sat, Oct 08, 2005 at 12:26:45PM -0400, Hal Ruhl wrote:
>
> 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].
>
I don't think this is right, but I could be grasping the wrong end of
the stick. I think of your definition division as the division of an infinite
length symbol string into a finite head, and a countably infinite long
tail. If true, then there are A^n heads of length n, and
c (=A^\aleph_0) tails.
Therefore, there are c pairs of descriptions.
Cheers
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Re: A question re measure
Hi Russell: At 12:33 AM 10/7/2005, you wrote: 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
Re: A question re measure
On Wed, Oct 05, 2005 at 10:45:28AM -0400, Hal Ruhl wrote:
> I am not a mathematician and so ask the following:
>
> 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.
>
> To me there should be far more highly asymmetric sized divisions
> [finite vs countably infinite] of the list than symmetric or nearly
> symmetric [countably infinite vs countably infinite] ones.
>
> However, for each small [finite] description there is a large
> [countably infinite] description.
>
> The result seems to be that there are more large descriptions than small
> ones.
>
> If the above is correct then mathematically what are the measures of
> the two types of descriptions?
>
> Hal Ruhl
>
>
>
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.
Cheers
--
*PS: A number of people ask me about the attachment to my email, which
is of type "application/pgp-signature". Don't worry, it is not a
virus. It is an electronic signature, that may be used to verify this
email came from me if you have PGP or GPG installed. Otherwise, you
may safely ignore this attachment.
A/Prof Russell Standish Phone 8308 3119 (mobile)
Mathematics0425 253119 (")
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