A generalization of Axiom A
Tadatoshi Miyamoto · Journal of the Mathematical Society of Japan · 1990
\S 1. Introduction.In [1], J. Baumgartner introduced the class of partial orderings for Axiom $A$ which includes $c.c.c$ .$p.0$ .sets, $\omega_{1}$ -closed $p.0$ .sets and various notions of forc- ing which add new subsets of $\omega$ .If partial orderings which satisfy Axiom $A$ are iterated under countable support, then the iteration, regardless of its length, satisfies the following covering property: If $X^{o}$ is a countable subset of the ordinals in the generic extension via the iteration, then there is $X\in V$ (the ground model) which is countable in $V$ with $X^{o}\subseteqq X$ .This covering property implies that $\omega_{1}$ is preserved.The main procedure involved in showing this is to produce what we call a fusion sequence which has a lower bound.It is not plausible, however, that the iteration itself satisfies Axiom $A$ .In this paper we generalize the class of partial orderings for Axiom $A$ so that our generalization is iterable under countable support.The difference be- tween these two classes is that: When we construct a nice descending sequence (fusion sequence) $\langle p_{n}\rangle_{n<\omega}$ , the choice of $p_{n+1}$ depends only on $p_{n}$ for Axiom $A$ and depends on $p_{0},$ $\cdots$ , $p_{n}$ for our generalization.Let us begin with a quick review of definitions.\S 2. Preliminaries.A binary relation $(P, \leqq)$ is a preordering if $(P, \leqq)$ is reflexive and transi- tive.A preordering $(P$ , $ $)$ satisfies Axiom $A$ if there is a sequence $\langle\leqq_{n}\rangle_{n<\omega}$ such that (1) $(P, \leqq_{n})$ is a preordering for all $n<\omega$ ,(2) if $p\leqq_{n}q$ , then $P\leqq q$ , (3) if $p\leqq_{n+1}q$ , then $p\leqq_{n}q$ , (4) if $\langle p_{n}\rangle_{n<\omega}$ is a sequence of conditions from $P$ with $p_{n+1}\leqq_{n}p_{n}$ for each $n<\omega$ , then there is a condition $P$ in $P$ such that $p\leqq_{n}p_{n}$ for all $n<\omega$ ,(5) for any $n$ in $\omega$ , any $p$ in $P$ and any dense subset $D$ of $P$ below $p$ (i.e.