The structure of Ext(A,Z) and GCH: possible co-Moore spaces

添加高度泛型的 $\omega_2$ 子集

Adding highly generic subsets of $ω_2$

本研究从GCH出发构造保基数与GCH的集合论泛型扩张，得到满足特定Cohen泛型性质的$\omega_2$子集，推进了无穷组合泛型集相关研究。

An Easton like theorem in the presence of Shelah Cardinals

We show that Shelah cardinals are preserved under the canonical $GCH$ forcing notion. We also show that if $GCH$ holds and $F:REG\rightarrow CARD$ is an Easton function which satisfies some weak properties, then there exists a cofinality preserving generic extension of the universe which preserves Shelah cardinals and satisfies $\forall κ\in REG,~ 2^κ=F(κ)$. This gives a partial answer to a question asked by Cody [1] and independently by Honzik [5]. We also prove an indestructibility result for Shelah cardinals.

Some results on large cardinals and the continuum function

Given a Woodin cardinal $δ$, I show that if $F$ is any Easton function with $F"δ\subseteqδ$ and $\GCH$ holds, then there is a cofinality-preserving forcing extension in which $2^γ= F(γ)$ for each regular cardinal $γ<δ$, and in which $δ$ remains Woodin. I also present a new example in which forcing a certain behavior of the continuum function on the regular cardinals, while preserving a given large cardinal, requires large cardinal strength beyond that of the original large cardinal under consideration. Specifically, I prove that the existence of a $λ$-supercompact cardinal $κ$ such that $\GCH$…

Telgarsky's conjecture may fail

Telgársky's conjecture states that for each $k \in \mathbb N$, there is a topological space $X_k$ such that in the Banach-Mazur game on $X_k$, the player {\scriptsize NONEMPTY} has a winning $(k+1)$-tactic but no winning $k$-tactic. We prove that this statement is consistently false. More specifically, we prove, assuming $\mathsf{GCH}+\square$, that if {\scriptsize NONEMPTY} has a winning strategy for the Banach-Mazur game on a $T_3$ space $X$, then she has a winning $2$-tactic. The proof uses a coding argument due to Galvin, whereby if $X$ has a $π$-base with certain nice properties, then {\s…

Collapsing the cardinals of $HOD$

Assuming that $GCH$ holds and $κ$ is $κ^{+3}$-supercompact, we construct a generic extension $W$ of $V$ in which $κ$ remains strongly inaccessible and $(α^+)^{HOD} < α^+$ for every infinite cardinal $α< κ$. In particular the rank-initial segment $W_κ$ is a model of ZFC in which $(α^+)^{HOD} < α^+$ for every infinite cardinal $α$.

Gödel--Löb 可证性逻辑在有限导出序数拓扑上的完备性

Completeness of the Gödel--Löb Provability Logic for Finite Derived Ordinal Topologies

本文在假设 GCH 和存在带相干测度序列的可测基数下，构造保持共尾性和 GCH 的力迫扩张，证明有限导出序数拓扑的 Gödel--Löb 逻辑完备性，并扩展 Beklemishev 定理至双模态逻辑。