arXiv subjects
Publications and source records attributed to Mohammad Golshani.
We show that if for any two elementary equivalent structures $\mathbf{M}, \mathbf{N}$ of size at most continuum in a countable language, $\mathbf{M}^ω/ \mathcal{U} \simeq \mathbf{N}^ω/ \mathcal{U}$ for some ultrafilter $\mathcal{U}$ on $ω,$ then $CH$ holds. We also provide some consistency results about Keisler and Shelah isomorphism theorems in the absence of $CH$.
Given a Fraïssé class $\mathcal{K}$ and an infinite cardinal $κ,$ we define a forcing notion which adds a structure of size $κ$ using elements of $\mathcal{K}$, which extends the Fraïssé construction in the case $κ=ω.$
Suppose $κ$ is a singular strong limit cardinal of countable cofinality and let $\langle κ_{n}: n<ω\rangle$ be an incrasing sequence of regular cardinals cofinal in $κ$. We show that if $cf(2^κ)= κ^+$, then forcing with the full product $\prod_{n<ω}Add(κ_n,1)$ collapses $2^κ$ into $κ^+$. This result gives a consistent positive answer to a question of Sy Friedman. We also give a new proof of a result due to Shelah by showing that if the sequence carries a scale of length $κ^+,$ then forcing with $\prod_{n<ω}Add(κ_n,1)$ adds a generic filter for $Add(κ^+, 1)$, and indeed \[ \prod_{n<ω}Add(κ_n,1)/fin \simeq Add(κ^+, 1). \]
Fix a ring $ R $ and look at the class of left $ R $-modules and naturally, we restrict ourselves to the case of rings such that this class is not too similar to the case $ R $ is a field. We shall solve Kaplansky test problems, all three of which say that we do not have decomposition theory, e.g., if the square of one module is isomorphic to the cube of another it does not follows that they are isomorphic. Our results are in the ordinary et theory, ZFC. For this, we look at bimodules, i.e., the structures which are simultaneously left $ R $-modules and right $ S $-modules with reasonable associativity, over a commutative ring $ T $ included in the centers of $ R $ and of $ S $. Eventually, we shall choose $ S $ to help solve each of the test questions. But first, we analyze what can be the smallest endomorphism ring of an $(R, S)$-bimodule. We construct such bimodules by using a simple version of the black box.
We discuss some properties of Cohen and random reals. We show that they belong to any definable partition regular family, and hence they satisfy most "largeness" properties studied in Ramsey theory. We determine their position in the Mahler's classification of the reals and using it, we get some information about Liouville numbers. We also show that they are wild in the sense of o-minimality, i.e., they define the set of integers.
Let $M$ denote the Merimovich's model in which for each infinite cardinal $λ, 2^λ=λ^{+3}$. We show that in $M$ the following hold: (1) Shelah's strong hypothesis fails at all singular cardinals, indeed, $\forall λ(λ$ is a singular cardinal $\Rightarrow pp(λ)=λ^{+3}).$ (2) For each singular cardinal $λ$ there is an inner model $N$ of $M$ such that $M$ and $N$ have the same bounded subsets of $λ,$ $λ$ is a singular cardinal in $N$, $(λ^{+i})^N=(λ^{+i})^M$, for $i=1,2,3,$ and $N \models 2^λ=λ^{+}$. Thus it is possible to add many new fresh subsets to $λ$ without adding any new bounded subsets to $λ$.
Starting from suitable large cardinals, we force the failure of (weak) diamond at the least inaccessible cardinal. The result improves an unpublished theorem of Woodin and a recent result of Ben-Neria, Garti and Hayut.
Assuming the existence of a strong cardinal $κ$, a weakly compact cardinal $λ$ above it and $γ> λ,$ we force a generic extension in which $κ$ is a singular strong limit cardinal of any given cofinality $δ$, $2^κ\geq γ$ and such that the tree property holds at $κ^{++}$.
We show that if $cf(2^{\aleph_0})=\aleph_1,$ then any non-trivial $\aleph_1$-closed forcing notion of size $\leq 2^{\aleph_0}$ is forcing equivalent to $Add(\aleph_1, 1),$ the Cohen forcing for adding a new Cohen subset of $ω_1.$ We also produce, relative to the existence of suitable large cardinals, a model of $ZFC$ in which $2^{\aleph_0}=\aleph_2$ and all $\aleph_1$-closed forcing notion of size $\leq 2^{\aleph_0}$ collapse $\aleph_2,$ and hence are forcing equivalent to $Add(\aleph_1, 1).$ These results answer a question of Scott Williams from 1978. We also extend a result of Todorcevic and Foreman-Magidor-Shelah by showing that it is consistent that every partial order which adds a new subset of $\aleph_2,$ collapses $\aleph_2$ or $\aleph_3.$
We discuss the generalized Kurepa hypothesis $KH_λ$ at singular cardinals $λ$. In particular, we answer questions of Erdös-Hajnal [1] and Todorcevic [6], [7] by showing that $GCH$ does not imply $KH_{\aleph_ω}$ nor the existence of a family $ \mathcal{F} \subseteq [\aleph_ω]^{\aleph_0}$ of size $\aleph_{ω+1}$ such that $\mathcal{F} \restriction X$ has size $\aleph_0$ for every $X \subseteq S, |X|=\aleph_0$.
We define a version of Gitik-Sharon diagonal Prikry forcing using a strongly compact cardinal, and prove its basic properties.
Assuming the existence of a proper class of supercompact cardinals, we force that for every regular cardinal $κ$, there are $κ^+$-Aronszajn trees and all such trees are special.
We give a detailed proof of the properties of the usual Prikry type forcing notion for turning a measurable cardinal into $\aleph_ω$.
Assuming the existence of a strong cardinal and a measurable cardinal above it, we construct a model of $ZFC$ in which for every singular cardinal $δ$, $δ$ is strong limit, $2^δ=δ^{+3}$ and the tree property at $δ^{++}$ holds. This answers a question of Friedman, Honzik and Stejskalova [8]. We also produce, relative to the existence of a strong cardinal and two measurable cardinals above it, a model of $ZFC$ in which the tree property holds at all regular even cardinals. The result answers questions of Friedman-Halilovic [5] and Friedman-Honzik [6].
We continue our study of the class $\mathscr{C}(D)$, where $D$ is a uniform ultrafilter on a cardinal $κ$ and $\mathscr{C}(D)$ is the class of all pairs $(θ_1, θ_2),$ where $(θ_1, θ_2)$ is the cofinality of a cut in $J^κ/D$ and $J$ is some $(θ_1+θ_2)^+$-saturated dense linear order. We give a combinatorial characterization of the class $\mathscr{C}(D)$. We also show that if $(θ_1, θ_2) \in \mathscr{C}(D)$ and $D$ is $\aleph_1$-complete or $θ_1 + θ_2 > 2^κ,$ then $θ_1=θ_2.$
We generalize the combinatorial principles $C_n(κ), C^s_n(κ)$ and $Princ(κ)$ introduced by various authors, and prove some of their properties and connections between them. We also answer a question asked by Juhasz-Kunen about the relation between these principles, by showing that $C_n(κ)$ does not imply $C_{n+1}(κ)$, for any $n>2$. We also show the consistency of $C(κ)+\neg C^s(κ).$
Assuming $κ$ is a supercompact cardinal and $λ$ is an inaccessible cardinal above it, we present an idea due to Magidor, to find a generic extension in which $κ=\aleph_ω$ and $λ=\aleph_{ω+1}.$
Assuming the existence of a strong cardinal $κ$ and a measurable cardinal above it, we force a generic extension in which $κ$ is a singular strong limit cardinal of any prescribed cofinality, and such that the tree property holds at $κ^{++}$.