We outline the proof of the consistency that there exists a saturated tail club guessing ideal on ω₁ which is not a restriction of the nonstationary ideal. A new class of forcing notions and the forcing axiom for the class are introduced for this purpose.

This paper will define a new cardinal called aStationary Cardinal. We will show that every weakly∏ 1 1 -indescribable cardinal is a stationary cardinal, every stationary cardinal is a greatly Mahlo cardinal and every stationary set of a stationary cardinal reflects. On the other hand, the existence of such a cardinal is independent of that of a∏ 1 1 -indescribable cardinal and the existence of a cardinal such that every stationary set reflects is also independent of that of a stationary (...) cardinal. As applications, we will show thatV=L implies ◊ κ 1 holds if κ is∏ 1 1 -indescribable and so forth. (shrink)

James Earl Baumgartner (March 23, 1943–December 28, 2011) came of age mathematically during the emergence of forcing as a fundamental technique of set theory, and his seminal research changed the way set theory is done. He made fundamental contributions to the development of forcing, to our understanding of uncountable orders, to the partition calculus, and to large cardinals and their ideals. He promulgated the use of logic such as absoluteness and elementary submodels to solve problems in set theory, he applied (...) his knowledge of set theory to a variety of areas in collaboration with other mathematicians, and he encouraged a community of mathematicians with engaging survey talks, enthusiastic discussions of open problems, and friendly mathematical conversations. (shrink)

We prove that the following two statements are equiconsistent: there exists a greatly Mahlo cardinal; there exists a regular uncountable cardinal κ such that no stationary subset of κ+ ∩ cof carries a partial square.

A stationary subset S of a regular uncountable cardinal κ reflects fully at regular cardinals if for every stationary set $T \subseteq \kappa$ of higher order consisting of regular cardinals there exists an α ∈ T such that S ∩ α is a stationary subset of α. Full Reflection states that every stationary set reflects fully at regular cardinals. We will prove that under a slightly weaker assumption than κ having the Mitchell order κ++ it is consistent that Full Reflection (...) holds at every λ ≤ κ and κ is measurable. (shrink)

We prove that it is consistent that there exists a saturated tail club guessing ideal on ω1 which is not a restriction of the non-stationary ideal. Two proofs are presented. The first one uses a new forcing axiom whose consistency can be proved from a supercompact cardinal. The resulting model can satisfy either CH or 20=2. The second one is a direct proof from a Woodin cardinal, which gives a witnessing model with CH.

We extract some properties of Mahlo’s operation and show that some other very natural operations share these properties. The weakly compact sets form a similar hierarchy as the stationary sets. The height of this hierarchy is a large cardinal property connected to saturation properties of the weakly compact ideal.

It is shown in this paper that it is consistent (relative to almost huge cardinals) for various club guessing ideals to be saturated.

The weakly compact reflection principle\\) states that \ is a weakly compact cardinal and every weakly compact subset of \ has a weakly compact proper initial segment. The weakly compact reflection principle at \ implies that \ is an \-weakly compact cardinal. In this article we show that the weakly compact reflection principle does not imply that \ is \\)-weakly compact. Moreover, we show that if the weakly compact reflection principle holds at \ then there is a forcing extension preserving (...) this in which \ is the least \-weakly compact cardinal. Along the way we generalize the well-known result which states that if \ is a regular cardinal then in any forcing extension by \-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if \ is a weakly compact cardinal then after forcing with a ‘typical’ Easton-support iteration of length \ the weakly compact ideal equals the ideal generated by the ground model weakly compact ideal. (shrink)

For n<ω, we say that theΠn1-reflection principle holds at κ and write Refln if and only if κ is a Πn1-indescribable cardinal and every Πn1-indescribable subset of κ has a Πn1-indescribable proper initial segment. The Πn1-reflection principle Refln generalizes a certain stationary reflection principle and implies that κ is Πn1-indescribable of order ω. We define a forcing which shows that the converse of this implication can be false in the case n=1; that is, we show that κ being Π11-indescribable of (...) order ω need not imply Refl1. Moreover, we prove that if κ is -weakly compact where α<κ+, then there is a forcing extension in which there is a weakly compact set W⊆κ having no weakly compact proper initial segment, the class of weakly compact cardinals is preserved and κ remains -weakly compact. We also formulate several open problems and highlight places in which standard arguments seem to break down. (shrink)

We show that the existence of a weakly compact cardinal over the Zermelo–Fraenkel's set theory ZF is proof-theoretically reducible to iterations of Mostowski collapsings and Mahlo operations.