Completion of mathematical systems
Arthur H. Kruse · Pacific Journal of Mathematics · 1962
l Introduction* The completion problem to be considered may be informally and tentatively described as follows.Let jy be a class of systems of some type (e.g., S^ will be the class of all fields in Example 1; cf.§ § 6,7).For all a,bes^ let "α < 6" mean that a is a subsystem (e.g., subfield in Example 1) of b.For each a e Ssf let π(a) be a set of propositional forms 1 involving unknowns (e.g., polynomial equations in one unknown in Example 1); each of these forms may become a true or false proposition upon substitution of elements of a for the unknowns; a substitution turning a form into a true proposition is a solution of the form.For each a e s/ let π\a) be the set of all members of π(a) with solutions (relative to a).If α, b e Szf and a <b, then each peπ(a) will correspond to some member, say pl(p), of π(b) (e.g., if sf is the class of all groups, the propositional form "y~λxy Φ x for some y in α" in unknown x could correspond to "y~λxy Φ x for some y in 6").We may say that a e s/ is complete if and only if for each b e s^ with a < b and each p e π(a): if p has no solution (relative to α), then p\{p) has no solution (relative to 6).(E.g., in Example 1, a field is complete if and only if it is algebraically closed.)The completion problem to be considered is: Does each a e s$f have a complete extension? 2 This extension problem will be formulated rigorously in § § 5,6.In some explicit special cases in modern algebra the existence of a complete extension rests on (transfinitely) recursive definitions the justification of which at first glance would seem to require a very strong version of the axiom of choice (cf.Remark 5 of § 7).In this paper the set-theoretic foundations of such procedures will be examined.The result is a theorem from which will follow the usual extension theorems via the usual weak version of the axiom of choice.2. Set'theoretic preliminaries* In axiomatic set theory one may consider the following versions of the axiom of choice.