Provable forms of Martin's axiom.
Gary P. Shannon · Notre Dame Journal of Formal Logic · 1990
It is shown that if Martin's Axiom (MA) is restricted to wellorderable sets, and if the countable antichain condition is replaced by either a finite antichain condition or a finite "Q-strong antichain" condition, then the resulting statements are forms of MA provable in ZF.Variations of these weak forms of MA are shown to be equivalent to the Axiom of Choice and some of its weak forms. DefinitionsLet (P,<) be a quasi-order, i.e., < is reflexive and transitive (if in addition < is antisymmetric, then (P,<) is a partial order).Two elements x, y of P are said to be incompatible if there does not exist zGPsuch that z ^ xand z ^y (otherwisexandy are said to be compatible).A subset / of P is said to be an antichain if any two elements of / are incompatible.c(x) denotes the set of elements of P that are compatible with x.A subset D of P is said to be dense if for all x G P there exists y G D such that y < x. *I would like to thank the referee for many helpful comments and suggestions.