# [FOM] Hanf's conjectures on finite axiomatizability

I. Natochdag natochdag at elsitio.net.uy
Mon Jul 26 18:22:06 EDT 2004

```Santiago Bazerque wrote on july the 8th:

"Does there exist a finitely axiomatizable undecidable theory with
countably many complete extensions?

Conjecture I. Every axiomatizable theory is isomorphic to a finitely
axiomatizable theory.

Conjecture II. Every finitely axiomatizable theory with countably many
complete extensions is isomorphic to a finitely axiomatizable theory
formulated with a finite number of unary predicates."

The paper "Boolean sentence algebras: Isomorphism constructions"(Journal
of symbolic logic), written jointly by Hanf and Myers Demonstrated:

- The possibility of "Axiomatizable maps" implying recursiveness, and
vice versa.
- If its syntax constructs unary relations a language is functional.
- Theories are isomorphic if relations of one are isomorphic to
relations of the other and one of them repeats itself.( it repeats
itself avoiding the possibility of theories with isomorphic relations
yet not isomorphic).

The method developed suffices to deduce some determinate form of
conjectures I-II: Conjecture I is deduced with the necessity of
the "repeating itself" relation. Conjecture II i see it deducible
from: "If its syntax constructs unary relations a language is
functional" and methods (developed in the paper) for representing
theories with finitely many relations in terms of unary relations. I'm
writing all these from memory ( i read the paper years ago) so possibly
the results are presented differently.
Hanf also wrote papers creating "non-recursive tilings of the plane",
and works postulating the least cardinal number implying that if a
theory constructs models beyond it a theory constructs uncountable
models (Shelah, Barwise and Friedman wrote brilliant papers and
conjectures for Hanf numbers). Re-thinking in Hanf's work evoked me a
few vague intuitions (maybe someone finds them useful):

- Relation of "the fundamental theorem of quantification theory" as
deduced for example by Smullyan in "First order logic" with conjecture
II. The possibility of completeness as a tautology and its relation to
incompleteness and undecidable theories, that is to say: deducing
completeness as a tautology implies deducing possible inconsistency;
deducing possible consistency implies deducing possible incompleteness:
Is possible a theory with these methods??.

This is my first posting so i greet you all,

Natochdag.

```