In this note, we show that the theorems in Z. Balogh [2] proved there under Axiom $R$ are already... more In this note, we show that the theorems in Z. Balogh [2] proved there under Axiom $R$ are already provable under Fodor-type Reflection Princi- ple (FRP) introduced in [9] or under a slight extension of FRP still much weaker than Axiom R.
We prove that the automorphism group of P(ω)/fin remains simple if ℵ 2 Cohen reals are added to a... more We prove that the automorphism group of P(ω)/fin remains simple if ℵ 2 Cohen reals are added to a model of ZFC + CH. 1 1 This paper is based on a part of author's Dissertation written under the supervision of Professor Sabine Koppelberg whom the author would like to thank for her encouragement during the preparation of the thesis. Thanks are also due to the referee for some valuable suggestions on improvement of the formulation.
A partial ordering P is said to have the weak Freese-Nation property (WFN) if there is a mapping ... more A partial ordering P is said to have the weak Freese-Nation property (WFN) if there is a mapping f : P → [P ] ≤ℵ 0 such that, for any a, b ∈ P , if a ≤ b then there exists c ∈ f (a) ∩ f (b) such that a ≤ c ≤ b. In this note, we study the WFN and some of its generalizations. Some features of the class of Boolean algebras with the WFN seem to be quite sensitive to additional axioms of set theory: e.g., under CH, every ccc complete Boolean algebra has this property while, under b ≥ ℵ 2 , there exists no complete Boolean algebra with the WFN. (Theorem 6.2).
We study combinatorial principles known as stick and club. Several variants of these principles a... more We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side-by-side product of partial orderings which we call pseudo-product. Using such products, we give several generic extensions where some of these principles hold together with ¬CH and Martin's Axiom for countable p.o.-sets. An iterative version of the pseudo-product is used under an inaccessible cardinal to show the consistency of the club principle for every stationary subset of limits of ω 1 together with ¬CH and Martin's Axiom for countable p.o.-sets.
We show that the notions of generic and Laver-generic supercompactness are first-order definable ... more We show that the notions of generic and Laver-generic supercompactness are first-order definable in the language of ZFC. This also holds for generic and Laver-generic (almost) hugeness as well as for generic versions of other large cardinals.
We give an equivalent, but simpler formulation of the axiom SEP introduced by Juhász and Kunen in... more We give an equivalent, but simpler formulation of the axiom SEP introduced by Juhász and Kunen in . Our formulation shows that many of the consequences of the weak Freese-Nation Property of P(ω) studied in already follow from SEP. We show that it is consistent that SEP holds while P(ω) fails to have the (ℵ 1 , ℵ 0 )-ideal property introduced in [2]. This answers a question addressed independently by Fuchino and by Kunen. We also consider some natural variants of SEP and show that certain changes in the definition of SEP do not lead to a different principle. This answers a question addressed by Blass.
Strong reflection principles with the reflection cardinal ≤ ℵ 1 or < 2 ℵ 0 imply that the size of... more Strong reflection principles with the reflection cardinal ≤ ℵ 1 or < 2 ℵ 0 imply that the size of the continuum is either ℵ 1 or ℵ 2 or very large. Thus, the stipulation, that a strong reflection principle should hold, seems to support the trichotomy on the possible size of the continuum. In this article, we examine the situation with the reflection principles and related notions of generic large cardinals.
Solovay's random-real forcing ([1]) is the standard way of producing real-valued measurable cardi... more Solovay's random-real forcing ([1]) is the standard way of producing real-valued measurable cardinals. Following questions of Fremlin, by giving a new construction, we show that there are combinatorial, measuretheoretic properties of Solovay's model that do not follow from the existence of real-valued measurability.
In , E.V. Ščepin gave a list of open problems on openly generated and some other related spaces. ... more In , E.V. Ščepin gave a list of open problems on openly generated and some other related spaces. In this note, we give solutions to the problems (3), ( ), ( ), ( ) of the list. For problems (3), ( ), we give negative solutions in ZFC. We show that the assertion of problem ( ) is independent from ZFC (under existence of some large cardinal). For problem (8) we give a negative answer under V = L. It is still open if also for this problem an independence result can be obtained similary to the problem (7). In the last section we mention some results connected to this question.
We introduce the notion of weakly extendible cardinals and show that these cardinals are characte... more We introduce the notion of weakly extendible cardinals and show that these cardinals are characterized in terms of weak compactness of second order logic. The consistency strength and largeness of weakly extendible cardinals are located strictly between that of strongly unfoldable (i.e. shrewd) cardinals, and strongly uplifting cardinals. Weak compactness of many other logics can be connected to certain variants of the notion of weakly extendible cardinals. We also show that, under V = L, a cardinal κ is the weak compactness number of L ℵ 0 ,II stat,κ,ω if and only if it is the weak compactness number of L II κ,ω . The latter condition is equivalent to the condition that κ is weakly extendible by the characterization mentioned above (this equivalence holds without the assumption of V = L).
In [FL88], Foreman and Laver proved, assuming the existence of a huge cardinal the consistency of... more In [FL88], Foreman and Laver proved, assuming the existence of a huge cardinal the consistency of the transfer property of maximal chromatic number from aleph_{2} to aleph_{1}. We combine this result with another result by Baumgartner [Bau84] to prove, also assuming the existence of a huge cardinal, that such a transfer property from aleph_{2} to aleph_{1} does not follow from the transfer property from aleph_{3} to aleph_{1}
We study the relationships between the properties of graphs: " of coloring number > $mu$&... more We study the relationships between the properties of graphs: " of coloring number > $mu$" and " of chromatic number > $mu$" for a regular cardinal $mu$ in terms of set-theoretic reflection of these properties. We show that under certain conditions the non-reflection of the property "of coloring number > $mu$" of graphs of bounded cardinality implies the non-reflection of the property "of chromatic number > $mu$" The implication is proved by interpolating it by non-reflection of the properties which are related to generalized and/or modified forms of Fodor-type Reflection Principle, Strong Chang' s Conjecture, Rado s Conjecture and Galvin' s Conjecture. As an application of this result we show a non reflection theorem on chromatic number > $mu$ which partially covers the results in Shelah [11]. Further results in this line will be presented in Fuchino, Ottenbreit and Sakai [9]
In this note, we show that the theorems in Z. Balogh [2] proved there under Axiom $R$ are already... more In this note, we show that the theorems in Z. Balogh [2] proved there under Axiom $R$ are already provable under Fodor-type Reflection Princi- ple (FRP) introduced in [9] or under a slight extension of FRP still much weaker than Axiom R.
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