FOM: Complex vs. Real, and stability

[Yoav]

In answer to Mark Steiner's question concerning the model therotic difference 
between C (the complex numbers) and R (the reals):
 
>"hierarchies" of logic that I'm familiar with cannot distinguish real
>from complex analysis, given that formally the only difference between
>them is that the complex numbers are ordered pairs of reals (or can be
>modeled that way).  Has any work been done to make these finer

  Formally C is NOT the same as ordered pairs of reals.  The latter model 
is the same as C together with the conjugation automormism \sigma, defined by 
\sigma(x + iy) = x - iy .  In both <R^2 ; 0,1,+,*> and <C ; 0,1,+,*,\sigma> one 
can define the predicate "z is real" - in the first by "z's second coordinate is 
0", in the second by "\sigma(z) = z".  From here one can get that these models 
are bi-interpretable.
  In C as a pure field (without \sigma in the language) this predicate cannot 
be defined (for example because Th(C), the theory of C, admits elimination of 
quantifiers, while Th(R) and the previous two theories do not).

  Moreover, Th(C) is what's called a 'stable' theory, meaning that it's very 
well-behaved, and can be investigated using the full power of Shelah's 
Classification theory.  
  On the other hand Th(R) interprets a linear order (define "x<y" by "there 
exists a non-zero z s.t. x + z^2 = y"), therefore is not stable.  For example 
one can use a 'small' set of parameters (the rationals Q) to define in Th(R) a 
'large' (2^[aleph_0]) set of types - a type for every 'standard' real (using 
dedekind cuts), the type of an 'infinite' real (it's consistent, i.e. finitely 
satisfiable ,to say "x is bigger than any rational"), the type of an 
'infinitesimal' real ("x is positive, but smaller than any positive rational"), 
etc.   
  
Name:         Yoav Yaffe
Occupation:   Ph.D. student in mathematics
Institution:  Hebrew University in Jerusalem, Israel
Interests:    The model theory of differential fields