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Boris,
See interposed
comments.
Regards,
Tim
Thanks for your list and this opportunity to discuss
Rosen's ideas!
Is it possible that my name (Boris
Saulnier) appears in the "from" email field, instead of
cybob10?
TG: This is a
function of your email client (or perhaps, your internet service provider if you
are using a web-based email). The list will use the name you send it. Right now,
your email client is sending "cybob10" as the name on your
emails.
I'm glad I can speak with you about this largest model
question as I think it is a keystone in Rosen argumentation.
I go on with Judith and Tim's answers to my first post.
*** (1) Models as sets or as mappings ***
Tim, concerning your first point, let's collect what
Rosen says:
a) "I shall call any _expression_ of a set S as a
cartesian product of quotient sets an analysis of S". (LI, p.162)
b) "I will call the mapping f:S -->X an observable of
S" (p.156)
c) "Every family of such observables gives rise to an
analysis" (p.164) (the initial set S is represented in terms of the
cartesian product of the spectra of observables. Let's remark here that the
observables mus be non linked).
d) "Every such analysis gives us a model"(p.164) : here
we might think that a model is an anlysis.
e) "The totality of analytic models can be identified
with the totality of equivalence relations on S, or the totality of sets of
observables" : here Rosen insists more on the observable side of the
model.
So it is not so clear if a model is defined as an
analysis or as a set of non linked observables. But I agree it should be
possible to go reversly from one to the other (this is the "trick" I was
speaking about in my first post).
Then Tim could you help me in having a better
understanding of the second part of your 1st point, when you say
:
>"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)."
My interpretation of this is that it will much more
fruitfull to see models as mappings, because it is the organization of
relations between all the models that will really count.
TG: Yes, I think we agree. My feeling is that models can be
represented as sets, but that representation is going to lose structure, in comparison to the representation of the
models as mappings. Going from mappings to sets is something like a forgetful
functor, in my view. For the purposes of Rosen's discussion of analytic vs.
synthetic models, the set representation is adequate, and it allows the
discussion to utilize the concepts of products and sums of
sets.
*** (2) Largest model of a mechanism
***
Let's come to the 2nd point.
I'm convinced by Tim's formulation, but we may still
have a step to clarify :
a) If the set of all (inequivalent) simulable
models Mi is countably infinite then we can list the countable infinite
sequence of finite programs Ui corresponding to these models, then
construct M' as the intersection of all the Mi's (related question : is this
infinite intersection also a model?), consider the associated program U', and
see that this U' must necessarily be (already) among the Ui's, so that we can
not have an countably infinite sequence of inequivalent models.
b) Then we need to proove that
"having no largest model" implies "having a countably infinite set of
inequivalent models".
That's easy, because all the models are simulable, and
because we have a 1-1 correspondance between models and programs, and because
the set of finite programs is countably infinite.
TG:
Agreed.
Nevertheless, don't you think we
still need to proove that the infinite intersection of models is still a
model?
TG: Hmmm.....I am
not sure. I think if we agree that the intersection of any finite number of
models is a model, then by induction so will the intersection of an infinite
number of models....at least when it is a countable infinity.
I will have to
think about this some more. At the moment, I cannot think of stronger
reasoning than that.
Boris
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