2 8 N ov 1 99 9 On versions of ♣ on cardinals larger than א 1 Mirna Džamonja

ar X iv : m at h / 96 11 20 9 v 1 [ m at h . L O ] 1 3 N ov 1 99 6 Chains of End Elementary Extensions of Models of Set Theory Andrés

Large cardinals arising from the existence of arbitrarily long end elementary extension chains over models of set theory are studied here. In particular, we show that the large cardinals obtained that way (‘Unfoldable cardinals’) behave as a ‘boundary’ between properties consistent with ‘V=L’ and existence of indiscernibles. We also provide an ‘embedding characterisation’ of the unfoldable card...

1 7 Se p 20 04 Diamond ( on the regulars ) can fail at any strongly unfoldable cardinal

If κ is any strongly unfoldable cardinal, then this is preserved in a forcing extension in which 3κ(reg) fails. This result continues the progression of the corresponding results for weakly compact cardinals, due to Woodin, and for indescribable cardinals, due to Hauser.

9 A ug 1 99 4 REMARKS ON א 1 - METRIZABLE NOT METRIZABLE

CWH,CWN stand for collectionwise Hausdorff and collectionwise normal respectively. We analyze the statement “there is a λ− CWH not CWH first countable (Hausdorff topological) space”. We prove the existence of such a space under various conditions, show its equivalence to: there is a λ-CWN not CWN first countable space and give an equivalent set theoretic statement; the nicest version we can obt...

/ 99 09 12 3 v 2 1 8 N ov 1 99 9 Correct Treatment of 1 ( η · k ) p Singularities in the Axial

More canonical forms and dense free subsets

Assuming the existence of ! compact cardinals in a model on GCH, we prove the consistency of some new canonization properties on א!. Our aim is to get as dense patterns in the distribution of indiscernibles as possible. We prove Theorem 2.1. Theorem 2.1. Suppose the consistency of “ZFC + GCH + there are in(nitely many compact cardinals”. Then the following is consistent: ZFC+GCH+ and for every ...