[FOM] Compactness, completeness and the axiom of choice

[jean-yves beziau]

The relation between the compactness theorem and 
the completeness theorem varies depending on specifities.
At the level of abstract logic (no specification of language), we have:
(1) Lindenbaum extension theorem: compactness 
theorem implies existence of a maximal extension
(2) Axiom of choice: equivalent to (1) as proved by Dzick
(3) Completeness is a corollary of (1) Â

I have presented a detailed study of this in my PhD:
Recherches sur la logique universelle, Dpt of 
Mathematics, University o Paris 7, 1995
and part of it has been published as
"La véritable portée du théorème de 
Lindenbaum-Asser", Logique & Analyse , 167-166 (1999), pp.341-359.
The situation is summarized in paricular in a diagram p.350

You can also have a look at:

David W. Miller
Some Restricted Lindenbaum Theorems Equivalent to the Axiom of Choice
Logica universalis 1 (2007), 183–199<
<http://www.springerlink.com/content/k62ml13t80g76v58/>http://www.springerlink.com/content/k62ml13t80g76v58/

In the book "completness theory for propositional 
logics" by W.Pogorzelski and P.Wojtylak
<http://www.springer.com/birkhauser/mathematics/book/978-3-7643-8517-0>http://www.springer.com/birkhauser/mathematics/book/978-3-7643-8517-0
you will find also many interesting results, in particular the following:

Metatheorem A.4. The following theorems are effectively equivalent:
(i) Stone's representation theorem for Boolean algebras.
(ii) Strong adequacy of the two element Boolean algebra (or matrix M2) for the
classical propositional logic.
(iii) Gödel Malcev's propositional theorem.
(iv) Structural completeness theorem for the classical propositional logic.
(v) Lindenbaum-Los's maximalization theorem.
(vi) Los' theorem on the representation of Lindenbaum-Tarski algebras.