[FOM] Theorem schemes

[John Baldwin]

On Wed, 21 Jun 2006 joeshipman at aol.com wrote:

> I am looking for examples of well-known mathematical results of the
> following form:
>
> (*) Whenever the sentences Phi_1, Phi_2, Phi_3,... are all true (in a
> structure satisfying certain other properties), the sentences Psi_1,
> Psi_2, Psi_3, ... are all true (in that same structure).
>
> An example would be my recent result improving the fundamental theorem
> of algebra, where the "certain other properties" are the axioms for
> fields, Phi_i is the statement "every polynomial whose degree is the
> ith prime number has a root" and Psi_j is the statement "every
> polynomial of degree j has a root".
>
If I understand correctly another example is:

usually phrased as: if a  polynomial map from C^n to C^n is one-to-one 
then it is onto.

In the suggested formalization

The phi_i are the axioms for an algebraically closed field of 
characteristic zero

and the psi_i are sentences asserting the proposition for polynomials in n 
variables and degree d (where i codes n and d).

See page 2 of Marker model theory of fields LN in Logic 5

or page 21-22 of Cherlin LNM 521 --- both published by Springer.

The proof (due to Ax) is a nice combination of compactness and the fact 
that for all 
there exist sentences are preserved by unions of chains.