Related papers: Diamonds
We investigate the problem of when $\leq\lambda$--support iterations of $<\lambda$--complete notions of forcing preserve $\lambda^+$. We isolate a property -- {\em properness over diamonds} -- that implies $\lambda^+$ is preserved and show…
We prove that, consistently, there exists a weakly but not strongly inaccessible cardinal $\lambda$ for which the sequence $\langle 2^\theta:\theta<\lambda\rangle$ is not eventually constant and the weak diamond fails at $\lambda$. We also…
Bounded stationary reflection at a cardinal $\lambda$ is the assertion that every stationary subset of $\lambda$ reflects but there is a stationary subset of $\lambda$ that does not reflect at arbitrarily high cofinalities. We produce a…
Under some cardinal arithmetic assumptions, we prove that every stationary subset of lambda of a right cofinality has the weak diamond. This is a strong negation of uniformization. We then deal with a weaker version of the weak diamond-…
If cf(kappa) = kappa, kappa^+< cf(lambda) = \lambda, then there is a stationary subset S of {delta<lambda:cf(delta)=kappa} in I[lambda]. Moreover, we can find <C_delta :delta in S>, C_delta a club of lambda, otp(C_delta)=kappa, guessing…
We strengthen the revised GCH theorem by showing, e.g., that for lambda=cf(lambda)>beth_omega, for all but finitely many regular kappa<beth_omega, lambda is accessible on cofinality kappa in a weak version of it holds. In particular,…
We prove the consistency of the failure of the weak diamond $\Phi_\lambda$ at strongly inaccessible cardinals. On the other hand, we show that the very weak diamond $\Psi_\lambda$ is equivalent to the statement $2^{<\lambda}<2^\lambda$ and…
This article continues Roslanowski and Shelah math.LO/9906024 and 1105.6049 We introduce here yet another property of (<lambda)-strategically complete forcing notions which implies that their lambda-support iterations do not collapse…
Our original aim was, in Abelian group theory to prove the consistency of: lambda is strong limit singular and for some properties of abelian groups which are relatives of being free, the compactness in singular fails. In fact this should…
This is a slightly corrected version of an old work. Under certain cardinal arithmetic assumptions, we prove that for every large enough regular $\lambda$ cardinal, for many regular $\kappa < \lambda$, many stationary subsets of $\lambda$…
We prove the consistency of: for suitable strongly inaccessible cardinal lambda the dominating number, i.e. the cofinality of lambda^lambda is strictly bigger than cov(meagre_lambda), i.e. the minimal number of nowhere dense subsets of…
We give two results on guessing unbounded subsets of lambda^+. The first is a positive result and applies to the situation of lambda regular and at least equal to aleph_3, while the second is a negative consistency result which applies to…
We introduce a new parametrized diamond principle denoted $\diamondsuit(\mathsf{LP})$. This principle is akin to the parametrized diamonds of Moore, Hru\v{s}\'ak, and D\v{z}amonja, each of which corresponds to some cardinal invariant of the…
The concept of jointness for guessing principles, specifically $\diamondsuit_\kappa$ and various Laver diamonds, is introduced. A family of guessing sequences is joint if the elements of any given sequence of targets may be simultaneously…
We prove that for any $C^1$-stably weakly shadowing transitive set $\Lambda$, either $\Lambda$ is a sink or a source, or $\Lambda$ admits a dominated splitting.
We investigate reflection of stationary sets in P_kappa lambda and prove a consistency result for the case when lambda is the successor of kappa.
Fix any $\lambda\in\mathbb{C}$. We say that a set $S\subseteq\mathbb{C}$ is $\lambda$-$convex$ if, whenever $a$ and $b$ are in $S$, the point $(1-\lambda)a+\lambda b$ is also in $S$. If $S$ is also (topologically) closed, then we say that…
It is well-known that the square principle $\square_\lambda$ entails the existence of a non-reflecting stationary subset of $\lambda^+$, whereas the weak square principle $\square^*_\lambda$ does not. Here we show that if…
In this paper we provide a general tool to prove the consistency of $I1(\lambda)$ with various combinatorial properties at $\lambda$ typical at settings with $2^\lambda>\lambda^+$, that does not need a profound knowledge of the forcing…
We will present a collection of guessing principles which have a similar relationship to $\diamond$ as cardinal invariants of the continuum have to $\CH$. The purpose is to provide a means for systematically analyzing $\diamond$ and its…