[FOM] Compactness, completeness and the axiom of choice
beziau100 at gmail.com
Sat Dec 25 19:54:47 EST 2010
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 vÃ©ritable portÃ©e du thÃ©orÃ¨me 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), 183199<
In the book "completness theory for propositional
logics" by W.Pogorzelski and P.Wojtylak
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) GÃ¶del 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.
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