[FOM] uncountable structures and core mathematics III: Lowenheim Skolem
John Baldwin
jbaldwin at uic.edu
Wed Feb 22 20:06:55 EST 2006
I remarked on necessity of using infinitary logic to study complex
exponentiation an earlier note in this series.
Eons ago Addison's metamath course at Berkeley included the topic:
The Lowenheim-Skolem-Godel-Malcev theorem (with a proof Tarski and Vaught)
This whimsy emphasized the roles of the upward and downward Lowenheim
Skolem theorem.
We might ask, does (C,+, X, exp) have an L omega_1 omega elementary
extension? It does if Zilber's conjectured axiomatization is correct.
Dave Marker verified for special 2-variable polynomials of one
Zilber's axioms schemes. (It needs to be shown for polynomials of all
arities)
Interesting work in
the theory of complex variables (Schanuel's conjecture, Hadamard
factorization) appears in Dave Marker's: A remark on Zilber's
psuedoexponentiation
http://www.math.uic.edu/~marker/eac-fin.pdf
In the next note in this series I will discuss Lowenheim-Skolem phenomena
in L_omega_1 omega without discussing links to core mathematics.
John T. Baldwin
Director, Office of Mathematics Education
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
312-413-2149
Room 327 Science and Engineering Offices (SEO)
851 S. Morgan
Chicago, IL 60607
Assistant to the director
Jan Nekola: 312-413-3750
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