Condensation-coherent global square systems

Canonical structure in the universe of set theory: part two

“…We note that the scale involved in the principle CS is a "Very Good Scale" in the sense of our paper [4]. This principle is closely related to some combinatorial principles of Donder et al [8] and Donder et al [7]. Definition 3.1.…”

The PCF conjecture and large cardinals

“…In L there exist tree-like continuous scales, even tree-like squared scales (see [5], the tree-like property comes just from condensation). In [9] Moti Gitik proved that it is possible to build a model without tree-like continuous scales using the Extender based forcing of [10,11], but it is not known if there exist infinite approachable sets in these models.…”

Weakly Compact Cardinals and Nonspecial Aronszajn Trees

“…(5) Since C is not trivialized by a club of k, there is no /c-branch through T(po). (3) is clear from the definition (2). (4) is Lemma 1.2 on p. 4 of [11]; further, by the coherence property of C, the RHS of the inequality of (4) is < card a.…”

“…We then proceed by choosing y', an immediate successor of y which does not occur in any of the previously chosen branches, and repeatedly applying (5) to the an which lie above \y'\, starting from y', as in Case 1. Note that f(y') = y, by (2), and that distinct pairs (i, y) give rise to distinct y' and therefore to distinct branches. Then, if w has been chosen as the top of the (i, y) branch, we let f(w) be any member of T\y\, distinct from y and all the f(w'), where w' is the top of a previously chosen branch.…”

“…Then, cf a = w, and Ca is taken to be any constructible, cofinal subset of a of order type uj. It is clear that (2) There is no stationary S Ç p of these "Case 1" ordinals such that for a < ß, both in S, ae C'0.…”