[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,


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