Same but multifaceted July 11, 2008Posted 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 and the field of complex numbers .
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 , its elementary theory, that is, the set of all closed first order formulae that are true in , has infinitely many models of cardinality continuum .
In naive terms, is rigid, while is soft and spongy and shape-shifting. However, has only trivial automorphisms (an easy exercise), while has huge automorphism group, of cardinality (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 , while 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 : 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 you take a maximal independent set of elements realizing the generic type, and take any permutation of , it extends to an automorphism of the model. So, if is of size , then any basis has size , and its automorphism group has size .
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 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.