Strong analogues of Martin's axiom Imply Axiom R
| Autor: | Robert E. Beaudoin |
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| Rok vydání: | 1987 |
| Předmět: | |
| Zdroj: | The Journal of Symbolic Logic. 52:216-218 |
| ISSN: | 1943-5886 0022-4812 |
| Popis: | We show that either PFA' or Martin's maximum implies Fleissner's Axiom R, a reflection principle for stationary subsets of PJ1G.). In fact, the "plus version" (for one term denoting a stationary set) of Martin's axiom for countably closed partial orders implies Axiom R. In [F] Fleissner introduces a stationary set reflection principle which he calls Axiom R. For any A ? w-)2 he calls a family T c PM2(A) tight if the union of any increasing chain from T of uncountable cofinality is also in T. Axiom R states that for any A ? W2 of uncountable cofinality, if S is stationary in P,1(A) and T is tight and unbounded in P,2(4), then there is an X E T such that stationarity of S reflects to X: S ri P. 1(X) is stationary in P. 1(X). Fleissner derived the consistency of this axiom by forcing over a model containing a supercompact cardinal. We shall show that Axiom R follows from either of PFA' or Martin's maximum. (Similar results on reflection principles following from Martin's maximum have been announced by Jech.) Actually, considerably less is needed to prove Axiom R. Let MA+(countably closed, K) be the following statement: If P is a countably closed partial order, D is a family of at most N, dense subsets of P, and {S,: a < K} is a family of cardinality K of P-terms, each forced by every condition in P to denote a stationary subset of w1, then there is a D-generic filter G on P so that for every a < K, S,(G) is stationary. Here S,(G) = {f3 < w1: 3p E G p c-"fl E Sa"}, the natural interpretation of the term Sa by the filter G. THEOREM 1. MA+(countably closed, 1) implies Axiom R. PROOF. Let A ? W2 have uncountable cofinality, let S c Pm 1(A) be stationary, and let T c PM22(A) be tight and unbounded. We must find X e Tsuch that S ri P.,1(X) is stationary in PJ1(X). We shall find X by applying MA+(countably closed, 1) to a notion of forcing that collapses the cardinality of A to N, but preserves stationarity of S. To insure that X will belong to T we force as follows. Let P be the set of all functions p such that dom(p) is a countable ordinal, ran(p) c T, and for any a |
| Databáze: | OpenAIRE |
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