[FOM] Theorem schemes
John Baldwin
jbaldwin at uic.edu
Wed Jun 21 17:36:14 EDT 2006
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.
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