# FOM: Complex vs. Real, and stability

Yoav yoavy at math.huji.ac.il
Wed Jan 26 11:31:28 EST 2000


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