[FOM] weak extension of ZFC

John Baldwin jbaldwin at uic.edu
Fri Dec 1 03:56:22 EST 2006

Has any thought been given towards `philosophical' justifications
for adding to ZFC the axiom E:

the exponential function is increasing

(for all $\lambda$, $2^\lambda < 2^{\lambda^+}$
or more strongly
if $\mu < \lambda$, $2^\mu < 2^\lambda$.)

That is are there arguments similar to the `iterative conception
of set' for why we should accept this principal?

One might think this was of just arithmetical interest. But axiom
E implies the Devlin Shelah weak diamond which implies Morley's
theorem for L_{omega_1, omega}.  (If a sentence of this logic is
categorical up to all aleph_omega it is categorical in all
cardinalities.)  Thus if one switches from first order to
infinitary logic a fundamental result of model theory requires a
slight extension of ZFC.

John T. Baldwin
Director, Office of Mathematics and Computer Education
Department of Mathematics, Statistics, and Computer Science
jbaldwin at uic.edu
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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