A class of strong diamond principles

Compactness and Guessing Principles in the Radin Extensions

We investigate the interaction between compactness principles and guessing principles in the Radin forcing extensions. In particular, we show that in any Radin forcing extension with respect to a measure sequence on $κ$, if $κ$ is weakly compact, then $\diamondsuit(κ)$ holds. This provides contrast with a well-known theorem of Woodin, who showed that in a certain Radin extension over a suitably prepared ground model relative to the existence of large cardinals, the diamond principle fails at a strongly inaccessible Mahlo cardinal. Refining the analysis of the Radin extensions, we consistently …

Diamonds, Compactness, and Measure Sequences

We establish the consistency of the failure of the diamond principle on a cardinal $κ$ which satisfies a strong simultaneous reflection property. The result is based on an analysis of Radin forcing, and further leads to a characterization of weak compactness of $κ$ in a Radin generic extension.

Weak prediction principles

We prove the consistency of the failure of the weak diamond $Φ_λ$ at strongly inaccessible cardinals. On the other hand, we show that the very weak diamond $Ψ_λ$ is equivalent to the statement $2^{<λ}<2^λ$ and hence holds at every strongly inaccessible cardinal.

可数集上的钻石原则与 Tukey-top 超滤

Diamond principles and Tukey-top ultrafilters on a countable set

本文在 ZFC 中构造了满足两类钻石猜测原则的 Tukey-top 超滤，证明完全猜测不可能且不存在 Dodd-sound 超滤，并利用这些原则构造 Tukey-top 的 q-点、证明相关超滤类在 Fubini 和下封闭，同时分离了两类猜测原则。

Selectively (a)-spaces from almost disjoint families are necessarily countable under a certain parametrized weak diamond principle

The second author has recently shown ([20]) that any selectively (a) almost disjoint family must have cardinality strictly less than $2^{\alpeh_0}$, so under the Continuum Hypothesis such a family is necessarily countable. However, it is also shown in the same paper that $2^{\alpeh_0} < 2^{\alpeh_1}$ alone does not avoid the existence of uncountable selectively (a) almost disjoint families. We show in this paper that a certain effective parametrized weak diamond principle is enough to ensure countability of the almost disjoint family in this context. We also discuss the deductive strength of t…

一个参数化的$\diamondsuit$用于Laver性质和$\mathcal P(ω)/\mathrm{Fin}$的非平凡自同构

A parametrized $\diamondsuit$ for the Laver property and nontrivial automorphisms of $\mathcal P(ω)/\mathrm{Fin}$

本文提出参数化$\diamondsuit(\mathsf{LP})$原理，证明其在特定forcing模型中成立，并展示其导致$\mathcal P(ω)/\mathrm{Fin}$存在非平凡自同构。

Prevalence of Generic Laver Diamond

Viale \cite{Viale_GuessingModel} introduced the notion of Generic Laver Diamond at $κ$---which we denote $\Diamond_{\text{Lav}}(κ)$---asserting the existence of a single function from $κ\to H_κ$ that behaves much like a supercompact Laver function, except with generic elementary embeddings rather than internal embeddings. Viale proved that the Proper Forcing Axiom (PFA) implies $\Diamond_{\text{Lav}}(ω_2)$. We strengthen his theorem by weakening the hypothesis to a statement strictly weaker than PFA. We also show that the principle $\Diamond_{\text{Lav}}(κ)$ provides a uniform, simple construc…

Positive provability logic for uniform reflection principles

We deal with the fragment of modal logic consisting of implications of formulas built up from the variables and the constant `true' by conjunction and diamonds only. The weaker language allows one to interpret the diamonds as the uniform reflection schemata in arithmetic, possibly of unrestricted logical complexity. We formulate an arithmetically complete calculus with modalities labeled by natural numbers and ω, where ωcorresponds to the full uniform reflection schema, whereas n<ωcorresponds to its restriction to arithmetical Π_{n+1}-formulas. This calculus is shown to be complete w.r.t. a su…

Joint Laver diamonds and grounded forcing axioms

I explore two separate topics: the concept of jointness for set-theoretic guessing principles, and the notion of grounded forcing axioms. A family of guessing sequences is said to be joint if all of its members can guess any given family of targets independently and simultaneously. I primarily investigate jointness in the case of various kinds of Laver diamonds. In the case of measurable cardinals I show that, while the assertions that there are joint families of Laver diamonds of a given length get strictly stronger with increasing length, they are all equiconsistent. This is contrasted with …

A new small Dowker space

It is proved that if there exists a Luzin set, or if either the stick principle or diamond(b) hold, then a strong instance of the guessing principle $\clubsuit_{AD}$ holds at the first uncountable cardinal. In particular, any of the above hypotheses entails the existence of a Dowker space of size $\aleph_1$.

来自$\clubsuit$型原理的非平凡导出极限

Nonvanishing derived limits from $\clubsuit$-type principles

本研究聚焦组合集合论中阿贝尔群逆系统的非平凡导出极限问题，改用$\clubsuit$型原理及其弱化形式开展探究，突破了以往依赖弱钻石类猜测原理的研究路径，拓展了相关非平凡性结果的推导工具范畴。

Continuous Maps on Aronszajn Trees

Assuming Jenson's principle diamond: Whenever B is a totally imperfect set of real numbers, there is special Aronszajn tree with no continuous order preserving map into B.