| Математическое---о теории моделей |
Apr. 13th, 2004|01:45 am |
Что такое теория моделвй ?
What is Model Theory? A theological essay.
0.1 Our Belief.
God has created the world so that man is able to learn
(no3HATb MUP) it. It is not any world
that can be learned by man. The very fact that man is
able to learn the world is of consequence.
In a world which can be learned, Free Will implies
the Law of Large Numbers. Thousands of free
choices create chaos; the Law of Large Numbers orders
the chaos so that we may understand it. If the Law of
Large Numbers fails, nothing remotely similar to the
world we know may exists. An innkeeper may not say
anything about his customers. Neither how many there
will be, nor what would they want to drink; Neither the
travellers would know where to expect an inn...Thus,
the World cannot be learned, from the point of view of
the innkeeper. (ideology behind theory of stability in model theory)
If there is no Free Will, there is no chaos; the
innkeeper perform well if he is supplying for the
troops; but, in this case, there is no chaos, and
everything is totally ordered. (ideology behind theory of o-minimality in model theory)
But, the Law of Large Numbers may hold only if there is
a good notion of Independence; and Independence implies
Probability Theory, and that is of consequence for the
World. Probability Theory describes precisely the
cities, the roads, and the width of the roads, and even
the golden mean;)....
The above argument is Model Theory. What is true in the
World is not for Model Theory---we do not care for
Probability Theory per se. We care for the Law of Large
Numbers only as much as it implies Probability Theory
and is implied by the learnability of the World and
Free Will. And for the Free Will we care because it is
a very simple structural assumption which allows us to
use the learnability of the World.;)
People learn the World through the Language. And the
argument above makes rather critical use of the Language
we use to describe the World in. Thus, in Model Theory,
the Language is very important. The notion of a
language has been formalised by Tarski, and that is the
formalisation we use in Model Theory. It is arguably
not the best and only one possible. One might perhaps
argue that Category Theory is also a notion of a
language; but in Model Theory, we do not even know what
Category Theory is, never use Categories (in fact,
there is no nice (useful) category associated a
theory)...Nor wish to know, for that matter.
People can learn the World through the Language. An
approximation to that is that the Language describes
the World completely. This can be formalised very
easily, almost in the same words: the theory of the
world (model) in the language describes it uniquely, up
to an isomorphism. This is a corner-stone of (modern)
Model Theory, called categoricity.
The above ideology sums in a slogan of Model Theory (in
fact, stability theory inside of Model theory).
The important objects of Mathematics are categorical
when considered in a proper language.
and its dual
The categorical objects are important in Mathematics.
More is true; there is a formal analogue that
Learnability implies The Law of Large Numbers; namely,
it is that
the categoricity does imply existence of an
independence relation between subsets (an independence
relation is understood as in van der Waerden).
Next section describes a couple of obvious examples.
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