arXiv subjects
Publications and source records attributed to Dima Sinapova.
In this paper we prove that the tree property can hold on regular cardinals in an interval which overlaps a strong limit cardinal. This is a crucial milestone in the long term project, tracing back to a question raised by Foreman and Magidor in the 1980s, of obtaining the tree property at every regular cardinal above the first uncountable cardinal.
In this paper we continue the study of equivalence of generics filters started by Smythe in [Smy22]. We fully characterize those forcing posets for which the corresponding equivalence of generics is smooth using the purely topological property of condensation. Next we leverage our characterization to show that there are non-homogeneous forcing for which equivalence of generics is not smooth. Then we prove hyperfiniteness in the case of Prikry forcing and some additional results addressing the problem whether generic equivalence for Cohen forcing is hyperfinite.
Combining stationary reflection (a compactness property) with the failure of SCH (an instance of non-compactness) has been a long-standing theme. We obtain this at $\aleph_{\omega_1}$, answering a question of Ben-Neria, Hayut, and Unger: We prove from the existence of uncountably many supercompact cardinals the consistency of $\aleph_{\omega_1}$ is strong limit together with $2^{\aleph_{\omega_1}}>\aleph_{\omega_1+1}$ and every stationary set of $\aleph_{\omega_1+1}$ reflects.
We prove the consistency of the failure of the singular cardinals hypothesis at $\aleph_ω$ together with the reflection of all stationary subsets of $\aleph_{ω+1}$. This shows that two classic results of Magidor (from 1977 and 1982) can hold simultaneously.
In Part I of this series, we introduced a class of notions of forcing which we call Sigma-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are Sigma-Prikry. We showed that given a Sigma-Prikry poset P and a P-name for a non-reflecting stationary set T, there exists a corresponding Sigma-Prikry poset that projects to P and kills the stationarity of T. In this paper, we develop a general scheme for iterating Sigma-Prikry posets and, as an application, we blow up the power of a countable limit of Laver-indestructible supercompact cardinals, and then iteratively kill all non-reflecting stationary subsets of its successor. This yields a model in which the singular cardinal hypothesis fails and simultaneous reflection of finite families of stationary sets holds.
We introduce a class of notions of forcing which we call $Σ$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $Σ$-Prikry. We show that given a $Σ$-Prikry poset $\mathbb P$ and a name for a non-reflecting stationary set $T$, there exists a corresponding $Σ$-Prikry poset that projects to $\mathbb P$ and kills the stationarity of $T$. Then, in a sequel to this paper, we develop an iteration scheme for $Σ$-Prikry posets. Putting the two works together, we obtain a proof of the following. Theorem. If $κ$ is the limit of a countable increasing sequence of supercompact cardinals, then there exists a cofinality-preserving forcing extension in which $κ$ remains a strong limit, every finite collection of stationary subsets of $κ^+$ reflects simultaneously, and $2^κ=κ^{++}$.
We use set-theoretic tools to make a model-theoretic contribution. In particular, we construct a \emph{single} $\mathcal{L}_{ω_1,ω}$-sentence $ψ$ that codes Kurepa trees to prove the consistency of the following: (1) The spectrum of $ψ$ is consistently equal to $[\aleph_0,\aleph_{ω_1}]$ and also consistently equal to $[\aleph_0,2^{\aleph_1})$, where $2^{\aleph_1}$ is weakly inaccessible. (2) The amalgamation spectrum of $ψ$ is consistently equal to $[\aleph_1,\aleph_{ω_1}]$ and $[\aleph_1,2^{\aleph_1})$, where again $2^{\aleph_1}$ is weakly inaccessible. This is the first example of an $\mathcal{L}_{ω_1,ω}$-sentence whose spectrum and amalgamation spectrum are consistently both right-open and right-closed. It also provides a positive answer to a question in [18]. (3) Consistently, $ψ$ has maximal models in finite, countable, and uncountable many cardinalities. This complements the examples given in [1] and [2] of sentences with maximal models in countably many cardinalities. (4) $2^{\aleph_0}<\aleph_{ω_1}<2^{\aleph_1}$ and there exists an $\mathcal{L}_{ω_1,ω}$-sentence with models in $\aleph_{ω_1}$, but no models in $2^{\aleph_1}$. This relates to a conjecture by Shelah that if $\aleph_{ω_1}<2^{\aleph_0}$, then any $\mathcal{L}_{ω_1,ω}$-sentence with a model of size $\aleph_{ω_1}$ also has a model of size $2^{\aleph_0}$. Our result proves that $2^{\aleph_0}$ can not be replaced by $2^{\aleph_1}$, even if $2^{\aleph_0}<\aleph_{ω_1}$.
For an inaccessible cardinal $κ$, the super tree property (ITP) at $κ$ holds if and only if $κ$ is supercomact. However, just like the tree property, it can hold at successor cardinals. We show that ITP holds at the successor of the limit of $ω$ many supercompact cardinals. Then we show that it can consistently hold at $\aleph_{ω+1}$. We also consider a stronger principle, ISP, and certain weaker variations of it. We determine which level of ISP can hold at a successor of a singular. These results fit in the broad program of testing how much compactness can exist in the universe, and obtaining large cardinal-type properties at smaller cardinals.