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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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