Mutual Stationarity
by
Philip Welch
University of Bristol

We survey some work around Foreman and Magidor's notion of mutual stationarity of a sequence of sets.

Definition A sequence S = áS_a | a < dñ with S_a Í k_a where d < k₀ < ¼ < k_b < k_{b+ 1} < ¼ is a sequence of regular cardinals, is called mutually stationary if every algebra A on supk_a has a subalgebra B Ì A satisfying: k_a Î |B| ® sup{|B| Çk_a} Î S_a.

Let cof_w1 = _df {a| a Î On Ùcf(a) = w₁}. In joint work with Peter Koepke we have shown the following two theorems relating independent choices of stationary sets independently to this notion of the sequence as a whole being mutually stationary:

Theorem 1 If every sequence of stationary sets S_n Ì k_n Çcof_w1 for n < w is mutually stationary then there is an inner model with sup{k_n} a measurable cardinal.

Theorem 2 If every sequence of stationary sets S_n Ì À_n Çcof_w1 for n < w is mutually stationary then there is an inner model with infinitely many measurable cardinals of Mitchell order o(w₁).

(It should be noted that it is not known to be consistent with ZFC whether every independently chosen sequence of stationary sets S_n Ì À_n Çcof_w1 for 1 < n < w, is mutually stationary. It is consistent by a theorem of Cummings-Foreman-Magidor that there is a sequence k_n for n < w for which every such choice of stationary sets S_n is mutually stationary, relative to the consistency of the existence of a measurable cardinal, thus rendering Theorem 1 an equiconsistency result.)

If one varies the cofinalities in the choice of the S_n sets larger cardinals are necessary. As sample:

Theorem 3 Suppose S_n = À_n Çw if n @ 0, 1 mod 4 and S_n = À_n Çw₁ otherwise. Then there is an inner model with a strong cardinal.

Date received: March 24, 2005