Pcf with Little Choice Sh955

Anotated Content §0 Introduction §1 On pseudo true cofinality, pg.2 [We continue [Sh:938, §5] to try to generalize the pcf theory for ℵ 1-complete filters D on Y assuming DC + AC P(Y). So is similar to [Sh:b, ChXII]. We suggest to replace cofinality by pseudo cofinality.] §2 Depth of reduced power of ordinals, pg.10 [Using the independence property for a sequence of filters we can bound the relevant depth.] 0. Introduction In the first section we deal with generalizing the pcf theory in the direction as started in [Sh:938, §5]. The point is that we assume AC U for any set U of power ≤ |P(P(Y))| or actually working harder, just ≤ |P(Y)| analyzing t∈Y α t , but we do not assume AC sup{αt:t∈Y }. The price is that we replace (true) cofinality by pseudo (true) cofinality. In the second section we prove a relative of [Sh:513, §3]; again dealing with depth (instead of rank as in [Sh:938]) adding some information even under ZFC. Assuming that the sequence D n : n < ω of filters has the independence property (IND), see Definition 2.3, with D n a filter on Y n we can bound the depth of (Yn) ζ by ζ for many m's, see 2.4. Of course, we can generalize this to D s : s ∈ S. This is incomparable with the results of [Sh:938, §4], also we add some cases to [Sh:938, §4]. Note that the assumptions like IND(¯ D) are complimentary to ones used in [Sh:835] to get considerable information. Our original hope was to arrive to a di-chotomy. The first possibility will say that one of the versions of an axiom suggested in [Sh:835] holds, which means " for some suitable algebra " , there is no independent ω-sequence; in this case [Sh:835] tells us much. The second possibility will be a