[FOM] Theorem schemes

joeshipman@aol.com joeshipman at aol.com
Wed Jun 21 03:28:10 EDT 2006

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".

In that case the Phi's are actually a subset of the Psi's, but I'm not 
requiring that property.

Another example involves a constant symbol. Roth's theorem can be 
approached by results in the form (*) as follows:

Fix a positive integer k (see discussion below for how high k has to 

Phi_i is the sentence

"c is not rational, and there exists a rational approximation p/q to c 
such that q>i and |c - (p/q)| < 1/(q^(2+(1/k)))".

Psi_j is the sentence

"c is not the root of a rational polynomial of degree j or less".

(*) here signifies that any irrational number with infinitely many good 
rational approximations is transcendental, where "good" is defined in 
terms of the exponent (2 + (1/k)).

Of course, Roth was actually able to show that the exponent in the 
approximation could be taken to be "2 plus epsilon", but Roth's theorem 
is notoriously nonconstructive. This means that there is already some k 
for which we don't have an algorithm for bounding the approximations a 
given algebraic number has with exponent (2+ 1/k); that's the k I have 
in mind above.

There's an interesting difference between these two examples. In my 
first example, the sentences Phi_i and Psi_j are adjoined to a finitely 
axiomatized first-order theory, and we can apply Godel's Compactness 
theorem to conclude that each Psi is implied by only finitely many of 
the Phi's. In the case of "degree axioms", I found an simply computable 
necessary and sufficient condition for whether a given set of Phi's 
sufficed for a given Psi.

In the second example, we are not dealing with a finitely axiomatized 
theory, and we also have to worry about nonstandard models. But is 
there still a result to be found of the form "each Psi depends on only 
finitely many of the Phi's"?

The reason I am looking for more examples of the scheme (*) is to find 
other cases of interesting finitary relationships between axioms 
belonging to an infinite set (in effect making Godel's Compactness 
Theorem constructive for those cases).

-- JS
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