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Model Theory and Category Theory July 29, 2008

Posted by dcorfield in Uncategorized.

David Kazhdan has some interesting things to say about model theory, and in particular its relationship to category theory, in his Lecture notes in Motivic Integration.

In spite of it successes, the Model theory did not enter into a “tool box” of mathematicians and even many of mathematicians working on “Motivic integrations” are content to use the results of logicians without understanding the details of the proofs.

I don’t know any mathematician who did not start as a logician and for whom it was “easy and natural” to learn the Model theory. Often the experience of learning of the Model theory is similar to the one of learning of Physics: for a [short] while everything is so simple and so easily reformulated in familiar terms that “there is nothing to learn” but suddenly one find himself in a place when Model theoreticians “jump from a tussock to a hummock” while we mathematicians don’t see where to “put a foot” and are at a complete loss.


Same but multifaceted July 11, 2008

Posted by Alexandre Borovik in Uncategorized.

Continuing the topic of “sameness”, it is interesting to compare behaviour of two familiar objects: the field of real numbers \mathbb{R} and the field of complex numbers \mathbb{C}.

\mathbb{C} is uncountably categorical, that is, it is uniquely described in a language of first order logic among the fields of the same cardinality.

In case of \mathbb{R}, its elementary theory, that is, the set of all closed first order formulae that are true in \mathbb{R}, has infinitely many models of cardinality continuum 2^{\aleph_0}.

In naive terms, \mathbb{C} is rigid, while \mathbb{R} is soft and spongy and shape-shifting. However, \mathbb{R} has only trivial automorphisms (an easy exercise), while \mathbb{C} has huge automorphism group, of cardinality 2^{2^{\aleph_0}} (this also follows with relative ease from basic properties of algebraically closed fields). In naive terms, this means that there is only one way to look at \mathbb{R}, while \mathbb{C} can be viewed from an incomprehensible variety of different point of view, most of them absolutely transcendental. Actually, there are just two comprehensible automorphisms of \mathbb{C}: the identity automorphism and complex conjugation. It looks like construction of all other automorphisms involves the Axiom of Choice. When one looks at what happens at model-theoretic level, it appears that “uniqueness” and “canonicity” of a uncountable structure is directly linked to its multifacetedness. I am still hunting appropriate references for this fact. Meanwhile, I got the following e-mail from a model theorist colleague, Zoe Chatzidakis:

Models of uncountably categorical theories behave really like vector spaces: if inside a model M you take a maximal independent set X of elements realizing the generic type, and take any permutation of X, it extends to an automorphism of the model. So, if M is of size \kappa > \aleph_0, then any basis has size \kappa, and its automorphism group has size 2^\kappa.

I don’t know a reference, but it should be in any model theory book which talks about strongly minimal sets. Or maybe in the paper by ??? Morley ??? which shows that you have a notion of dimension and so on? I.e., that \aleph_1 categorical theories and strongly minimal sets are the same.

It is really a well-known result, so you probably don’t need a reference if you cite it in a paper.

Facing Eternity July 4, 2008

Posted by Alexandre Borovik in Uncategorized.
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Mikhail Zlatkovsky. Facing Eternity.

Under a changed title “Coat Star” the cartoon won first prize at Ken Sprague Fund competition at  Earthworks 2008.