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Mohammad Golshani

Publications and source records attributed to Mohammad Golshani.

At least 37 records · Page 2Linked to original sources
arXiv2024

Cohen Generic Structures with Functions

Suppose $\mathscr{L}^-\subseteq \mathscr{L}$ are languages where $\mathscr{L} \setminus\mathscr{L}^-$ is relational. Additionally, let $\mathbf{K}$ be a strong $\textrm{Fraïssé}$ class in $\mathscr{L}$. We consider the partial ordering, under substructure, of those elements in $\mathbf{K}$ whose reduct to $\mathscr{L}^-$ are substructures of a fixed $\mathscr{L}^-$-structure $\mathcal{M}^-$. In this paper, we establish that, under general conditions, this partial order satisfies the $|\mathcal{M}^-|$-chain condition. Furthermore, under these conditions, we demonstrate that any generic for such a partial order satisfies the theory of the \Fraisse\ limit of $\mathbf{K}$, provided $\mathcal{M}^-$ satisfies the theory of$\textrm{Fraïssé}$ limit of its age. We also provide general conditions that guarantee all such generics to be rigid, as well as conditions ensuring that these generics possess large automorphism groups.

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arXiv2024

The special Aronszajn tree property at $\aleph_2$ and $GCH$

Starting from the existence of a weakly compact cardinal, we build a generic extension of the universe in which $GCH$ holds and all $\aleph_2$-Aronszajn trees are special and hence there are no $\aleph_2$-Souslin trees. This result answers a well-known open question from the 1970's.

math.LO↗
arXiv2023

Co-Hopfian and boundedly endo-rigid mixed abelian groups

For a given cardinal $λ$ and a torsion abelian group $K$ of cardinality less than $λ$, we present, under some mild conditions (for example $λ=λ^{\aleph_0}$), boundedly endo-rigid abelian group $G$ of cardinality $λ$ with $Tor(G)=K$. Essentially, we give a complete characterization of such pairs $(K, λ)$. Among other things, we use a twofold version of the black box. We present an application of the construction of boundedly endo-rigid abelian groups. Namely, we turn to the existing problem of co-Hopfian abelian groups of a given size, and present some new classes of them, mainly in the case of mixed abelian groups. In particular, we give useful criteria to detect when a boundedly endo-rigid abelian group is co-Hopfian and completely determine cardinals $λ> 2^{\aleph_{0}}$ for which there is a co-Hopfian abelian group of size $λ$.

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arXiv2023

Adding highly undefinable sets over $L$

We sow that there exists a generic extension of the Gödel's constructible universe in which diamond holds and there exists a subset $Y \subseteq ω_1$ such that for stationary many $δ< ω_1,$ the set $Y \cap δ$ is not definable in the structure $(L_{F(δ)}, \in)$, where $F(δ) > δ$ is the least ordinal such that $L_{F(δ)}\models$``$δ$ is countable''.

math.LO↗
arXiv2023

Definable tree property for uncountable regular cardinals

The primary goal of this paper is to establish a model of $ZFC$ wherein the definable tree property is affirmed for all uncountable regular cardinals. This endeavor commences with the utilization of both a supercompact cardinal and a measurable cardinal that exceeds it. Subsequently, we construct a $ZFC$ model. Within this model, we demonstrate that the definable tree property holds for all uncountable regular cardinals. Thereby we respond to an inquiry raised in [1].

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arXiv2023

Adding highly generic subsets of $ω_2$

Starting from the $\rm{GCH},$ we build a cardinal and $\rm{GCH}$ preserving generic extension of the universe, in which there exists a set $A \subseteq ω_2$ of size $\aleph_2$ so that every countably infinite subset of $A$ or $ω_2 \setminus A$ is Cohen generic over the ground model.

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arXiv2023

Unlimited accumulation by the Shelah's PCF operator

Modulo the existence of large cardinals, there is a model of set theory in which for some set $B$ of regular cardinals, the sequence $\langle \text{pcf}^α(B): α\in \text{Ord} \rangle$ is strictly increasing. The result answers a question from [7].

math.LO↗
arXiv2023

On slow minimal reals I

Answering a question of Harrington, we show that there exists a proper forcing notion, which adds a minimal real $η\in \prod_{i<ω} n^*_i$, which is eventually different from any old real in $\prod_{i<ω} n^*_i$, where the sequence $\langle n^*_i \mid i<ω\rangle$ grows slowly.

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arXiv2023

On the classification of definable ccc forcing notions

We show that for a Suslin ccc forcing notion $\mathbb Q$ adding a Hechler real, ``$\text{ZF}+\text{DC}_{ω_1}+$all sets of reals are $I_{\mathbb Q,\aleph_0}$-measurable'' implies the existence of an inner model with a measurable cardinal. We also introduce a wide class of Suslin ccc forcing notions which add a Hechler real, so that the above result applies to them.

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arXiv2022

The Keisler-Shelah isomorphism theorem and the continuum hypothesis II

We continue the investigation started in [Sh:1215] about the relation between the Keilser-Shelah isomorphism theorem and the continuum hypothesis. In particular, we show it is consistent that the continuum hypothesis fails and for any given sequence $\mathbf m=\langle (\mathbb{M}^{1}_n, \mathbb{M}^{2}_n: n < ω\rangle$ of models of size at most $\aleph_1$ in a countable language, if the sequence satisfies a mild extra property, then for every non-principal ultrafilter $\mathcal D$ on $ω$, if the ultraproducts $\prod\limits_{\mathcal D} \mathbb{M}^{1}_n$ and $\prod\limits_{\mathcal D} \mathbb{M}^{2}_n$ are elementarily equivalent, then they are isomorphic.

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arXiv2022

Graphs represented by Ext

This paper opens and discusses the question originally due to Daniel Herden, who asked for which graph $(μ,R)$ we can find a family $\{\mathbb G_α: α< μ\}$ of abelian groups such that for each $α,β\inμ$: $$Ext(\mathbb G_α, \mathbb G_β) = 0 \Longleftrightarrow(α,β) \in R.$$ In this regard, we present four results. First, we give a connection to Quillen's small object argument which helps $Ext$ vanishes and uses to present useful criteria to the question. Suppose $λ= λ^{\aleph_0}$ and $μ= 2^λ$. We apply Jensen's diamond principle along with the criteria to present $λ$-free abelian groups representing bipartite graphs. Third, we use a version of the black box to construct in ZFC, a family of $\aleph_1$-free abelian groups representing bipartite graphs. Finally, applying forcing techniques, we present a consistent positive answer for general graphs.

math.AC↗
arXiv2022

NNR Revisited

We show if we use countable support iteration of forcing notions not adding reals that satisfy additional conditions, then the limit forcing does not add reals. As a result we prove that we can amalgamate two earlier methods and prove the consistency with ZFC + GCH of two statements gotten separately earlier: Souslin hypothesis and non-club guessing. We also answer a question of Justin Moore by proving the consistency of one further case of "strong failure of club guessing" with GCH.

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