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Hi
Boris,
Welcome to the
list. :)
1) I disagree that
"models are sets (cartesian products or direct sums)". Instead, the
totality of analytical models of a system arises from
a Cartesian product (of the spectra of the family of observables)[LI 164],
and similarly, the totality of synthetic models of a
system arises from a corresponding direct sum. Each model (analytic
or synthetic) will have relations between observables which will place that
model as members in the set of relations specified by a member of
either the direct product or the direct sum (or, of course,
where direct product/direct sum overlap, of both).
2) The argument on
LI 205 regarding the largest model for a mechanism. I believe this argument is a
kind of Cantor-like diagonal argument. He begins by hypothesizing the
negative: that there is no largest model for a mechanism and therefore there are
a countably infinite number of inequivalent simulable models. Each Mi
thus has a corresponding and unique program associated with it, since each Mi is
simulable by hypothesis. Then, he proposes taking the intersection of all
the Mi, which will result in yet another M' which must be inequivalent
to all the other Mi. And this M' must also have a corresponding and unique
program (again, because by hypothesis, all these models are simulable). But, if
the set of Mi is already a countably infinite set, then M' must already a
member of {Mi}, and therefore the program for M' must also be a member of the
corresponding set of programs for {Mi}. Therefore M' cannot be
distinct, nor can the program for M' be distinct, and therefore the
hypothesis fails, and the opposite must be true: there is a largest model for a
mechanism.
Regards,
Tim
I'd like to submit to Rosen readers what might
appear at first sight as a rather technical "detail". But it is not only a
math problem because it is the difference between relational models and
(analytic=synthetic) models that is put into question.
I shall begin with 2 remarks, and then come to my
point.
Remark 1: in LI, p.203, section 8B (Machines and
mechanisms), Rosen says "A natural system N is a mechanism if and only if all
of its models are simulable". But from what I
understoood, the notion of "simulability" (calculability) covers mappings.
Whereas models are sets (cartesian products or direct sums). As a consequence
the notion of simulability for a model is not straightforward.
However, it might be possible to see a set
A=\prod Ai (cartesian product of sets Ai), that is to say an analytic
model, as the identity mapping from to A to A, and say that A is
simulable if and only if this identity mapping is simulable. And in the case
where A is a synthetic model just consider the associated analytic model, and
apply the same "trick". So it should be ok.
Remark 2: p.205, section 8C (On the largest
model of a mechanism), we want to prove that if a natural system N is a
mechanism then it has a unique largest model. First we suppose that the
category C(N) of all models of N has no largest model. Rosen says "Thus we can find an infinite sequence of
increasingly refined models M1<M2<...<Mi<...". This is not
obvious, but seems to be true, if you give yourself the "choice axiom" (so
called in french) of set theory, and consider that it is always possible to construct the refinement of two
models.
But then, and here I come to my point, Rosen
says "We thus end up with a countable family of distinct programs, each of
which is a distinct word of finite length on a finite alphabet : this is
clearly impossible". This is where I don't understand. Consider for
instance the sequence of all binary programs (binary sequences)
ordered in lexicographic order
(0<1<00<01<10<10<000<001<010<010<011<100<....) :
this is a countable family of distinct programs, each of which is a distinct
word of finite length on a finite alphabet. So it seems possible to have such
a family.
Am I mistaking? If not, we loose the property
that the category C(N) of all models of a mechanism has a largest model, and
we loose the property that if N is a mechanism then analytic and synthetic
coincide in C(N) (conclusion 4, p.205, section 8B).
What do you think?
Boris
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