Boolean algebras, partial orders and axiom of choice
Abstract
The objective of this paper is to present a demonstration of a classical theorem on boolean algebras and partial orders of current relevance in set theory, as for example, for applications of model construction method called “forcing” (with boolean algebras complete or with partial orders). The theorem to be proved is as follows: “Any partial order can be extended to a single complete boolean algebra (up to isomorphism)". Where to extend means “embed densely”. Such a demonstration is done using Dedekind's cuts following the text “Set Theory” of Jech, and other ideas of the author of this article. In addition, some weak versions of the axiom of choice related to boolean algebras are formulated, which are also of great importance for the research in set theory and model theory, since this are powerful model construction techniques, such as the compactness theorem (allows the construction of non-standard models, etc.) and the ultrafilter theorem, which allows the construction of ultraproducts (can be used to investigate problems of large cardinals, etc). Some references of open problems on the subject are presented.
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