arXiv subjects

Mohammad Golshani

Publications and source records attributed to Mohammad Golshani.

At least 19 recordsLinked to original sources

arXiv2026

A Set-Theoretic Translation of Modal Logic via Forcing

We develop a set-theoretic translation of a normal modal extension $T_m$ of a recursively axiomatizable first-order theory $T$. We first pass to the Henkin expansion of the underlying language by adding witness constants and work with the sentence algebra of this expansion. The translation is constructed using the corresponding Lindenbaum-Tarski algebra, the Stone space of its ultrafilters, and quotient forcing by a $σ$-ideal of Borel sets; in particular, the meager ideal yields Cohen forcing and the null ideal yields the random forcing. In a Boolean-valued universe, we interpret the modal operator $\Box$ by membership of the Boolean value of the translated formula in a suitable filter name. We prove that this translation is sound and complete: a modal formula is provable in $T_m$ if and only if its set-theoretic translation is forced in every associated interpretation. We then build Kripke frames and models from generic extensions and show that the forcing interpretation of $\Box$ agrees with quantification over the corresponding accessibility relation. As a consequence, we obtain completeness with respect to the resulting Kripke models.

math.LO↗

arXiv2026

Representing a ring as the endomorphism ring of an abelian group

We investigate Baer's realization problem for almost free abelian groups, focusing on the extent to which rings can be represented as endomorphism rings under strong freeness conditions. Building on earlier work that relied on additional set-theoretic principles such as the diamond or strong black boxes, we develop new methods that significantly weaken these assumptions. The main result is obtained in ZFC (assuming a mild cardinal arithmetic configuration): for a strong limit singular cardinal $μ$ with $μ^+ < 2^μ< 2^{μ^+}$, and for a wide class of cotorsion-free rings, we construct $μ^+$-free modules whose endomorphism rings are isomorphic to the given ring. This provides a substantial partial solution to a problem of Göbel and Trlifaj. The key innovation is the integration of $κ$-frame constructions with Shelah's Super Black Box, enabling a delicate diagonalization that eliminates nontrivial endomorphisms while preserving high degrees of freeness.

math.LO↗

arXiv2026

Cohen Forcing with Fraïssé Classes Beyond $\aleph_1$

Kostana [5] asked for natural conditions on a class $\mathcal{K}$ to ensure that forcing with finite members of $\mathcal{K}$ on an underlying set $X$ is forcing equivalent to Cohen forcing. We show that if $\mathcal{K}_{\triangle}$ is the class of finite triangle-free graphs and $|X|\geq\aleph_2$, then $ \mathbb{C}_X(\mathcal{K}_{\triangle}) \not\simeq \mathbb{C}[X]. $ We then use a characterization of Cohen algebras due to Milovich to obtain a positive result. We show that if $\aleph_n\leq|X|$ and $\mathcal{K}$ has $(n+1)$-disjoint amalgamation, then $\mathbb{C}_X(\mathcal{K})$ is Cohen.

math.LO↗

arXiv2026

Martin's axiom and $ω_1^2 \longrightarrow (ω_1^2, 3)^2$

Starting with CH and Hajnal's coloring, we show that a standard finite-support iteration of $σ$-centered forcing notions gives a model of $ MA_{ω_1}(σ$-centered$)+2^{\aleph_0}=\aleph_2 +ω_1^2\nrightarrow(ω_1^2,3)^2.$ We also isolate a simple forcing-preservation principle: the same ground-model coloring remains a witness after forcing with any poset whose subfamilies of size at most $ω_1$ are countable unions of linked sets. Under $\text{MA}_{ω_1}$, every c.c.c. forcing has this local property, so every existing witness is preserved by every c.c.c. forcing over that model.

math.LO↗

arXiv2026

Iterated Ramsey bounds for the Hales-Jewett numbers

Consider the Hales-Jewett theorem. The $k$-dimensional version of it tells us that the combinatorial space $\mathcal{U}_{M, Λ} = \{ η\mid η: M \to Λ\}$ has, under suitable assumptions, monochromatic $k$-dimensional subspaces, where by a $k$-dimensional subspace we mean there exist a partition $\langle N_0, N_1, \cdots, N_k \rangle$ of $M$ such that $N_1, \cdots, N_k \neq \emptyset$ (but we allow $N_0$ to be empty) and some $ρ_0: N_0 \to Λ$, such that the subspace consists of those $ρ\in \mathcal{U}_{M, Λ}$ such that for $0<l<k+1, ρ\restriction N_l$ is constant and $ρ\restriction N_0= ρ_0.$ It seems natural to think it is better to have each $N_{l}, 0<l<k+1$ a singleton. However it is then impossible to always find monochromatic $k$-dimensional subspaces (for example color $η$ by $0$ if $|η^{-1}\{α\}|$ is an even number and by $1$ otherwise). But modulo restricting the sign of each $|η^{-1}\{α\}|$, we prove the parallel theorem -- whose proof is not related to the Hales-Jewett theorem. We then connect the two numbers by showing that the Hales-Jewett numbers are not too much above the present ones. This gives an alternative proof of the Hales-Jewett theorem.

math.CO↗

arXiv2026

From Shelah's block-content to Hales-Jewett

We study the quantitative relationship between the Hales-Jewett numbers and Shelah's block-content canonization functions. Block-content canonization yields a block subspace on which the color of a word is determined solely by the multiplicities of the alphabet letters among the variable blocks. We show that this canonical information, combined with the multidimensional Gallai-Witt theorem, suffices to produce a monochromatic Hales-Jewett subspace. The argument passes to the space of content vectors, finds a monochromatic homothetic copy of a finite content simplex, and lifts it through the canonical block subspace. Combined with the elementary fact that Hales-Jewett bounds block canonization, this gives a two-way quantitative comparison up to an explicit change of parameters. The underlying mechanism may be summarized by the slogan Hales-Jewett = block canonization + Gallai-Witt We also prove the corresponding equal-block result and show, by an explicit coloring over every finite field of odd prime order, that the analogous singleton-coordinate canonization principle fails as soon as two coordinates remain live.

math.CO↗

arXiv2026

Forcing with Invariant Measures

This paper introduces a model-theoretic generalization of the notion of forcing with random reals, in which forcing gives rise to random generic structures. Specifically, we consider forcing with $κ$-Borel probability measures on the space of $L$-structures with a (possibly uncountable) infinite set $X$, focusing on those that are invariant under the action of the symmetric group $Sym(X)$. We demonstrate how any $Sym(X)$-invariant measure where $X$ is countable can be uniquely extended to a $Sym(Y)$-invariant measure where $Y$ is uncountable, and prove that forcing with such measures satisfies the countable chain condition. We also show that we can uniformly distinguish between these random generic structures and the Cohen generic structures that arise from forcing with a strong Fraïssé class: There is a $κ$-Borel set of low complexity that contains every Cohen generic structure that is not highly homogeneous but contains no random generic structure, implying that a structure that is not highly homogeneous cannot be both Cohen generic and random generic. Finally, we answer an open question of Kostana in the case of $ω_1$, by establishing a connection between forcing with a strong Fraïssé class and Cohen forcing.

math.LO↗

arXiv2026

Strong Completeness of Provability Logic for Uncountable Languages

For an ordinal $λ>0$, we use the Erdős--Rado partition theorem to prove the failure of strong completeness of $\mathsf{GL}$ for modal languages of cardinality $(2^{|λ|+\aleph_0})^{+}$ with respect to models on ordinals equipped with the generalized Icard topologies $\mathcal{I}_λ$ and ${τ_{c}}_{+λ}$. Specifically, we show that for such languages there exists a $\mathsf{GL}$-consistent set of formulas having neither $(Θ, \mathcal{I}_λ)$-model nor $(Θ, {τ_{c}}_{+λ})$-model. We also introduce two kinds of natural classes of topological spaces, called \emph{ $λ$-bouquet spaces} and \emph{ultralinear $λ$-bouquet spaces}, and prove that they yield strong completeness of $\mathsf{GL}$ and $\mathsf{GL}.3$ respectively for languages of cardinality $λ$.

math.LO↗

arXiv2026

Failure of singular compactness for Hom

Assuming Gödel's axiom of constructibility $V=L$, we construct a $χ$-free abelian group $G$ of singular cardinality for some suitable cardinal $χ$ which is regular and uncountable, equipped with the property that for every nontrivial subgroup $G' \subseteq G$ of smaller cardinality, $Hom(G',\mathbb{Z}) \neq 0$, while $Hom(G,\mathbb{Z}) = 0$. This provides a consistent counterexample to the singular compactness of nontrivial duality with respect to the functor $Hom(-,\mathbb{Z})$.

math.GR↗

arXiv2025

Quite free p-groups with trivial duality

We present a class of abelian groups that exhibit a high degree of freeness while possessing no non-trivial homomorphisms to a canonical free object. Unlike prior investigations, which primarily focused on torsion-free groups, our work broadens the scope to include groups with torsion. Our main focus is on p-groups, for which we formulate and prove the Trivial Duality Conjecture. Key tools in our analysis include the multi black box method and the application of specific homological properties of relative trees.

math.GR↗

arXiv2025

Naturality and Definability III

In this paper, we deal with the notions of naturality from category theory and definablity from model theory and their interactions. In this regard, we present three results. First, we show, under some mild conditions, that naturality implies definablity. Second, by using the reverse Easton iteration of Cohen forcing notions, we construct a transitive model of ZFC in which every uniformisable construction is weakly natural. Finally, we show that if F is a natural construction on a class K of structures which is represented by some formula, then it is uniformly definable without any extra parameters. Our results answer some questions by Hodges and Shelah.

math.LO↗

arXiv2025

The proper forcing axiom for $\aleph_1$-sized posets, $ω_1$-linked symmetrically proper forcing, and the size of the continuum

We show that the Proper Forcing Axiom for forcing notions of size $\aleph_1$ is consistent with the continuum being arbitrarily large. In fact, assuming $GCH$ holds and $κ\geqω_2$ is a regular cardinal, we prove that there is a proper and $\aleph_2$-c.c.\ forcing giving rise to a model of this forcing axiom together with $2^{\aleph_0}=κ$ and which, in addition, satisfies all statements of the form $\mathcal{H}(\aleph_2)\models \exists yφ(a, y)$, where $a\in \mathcal{H}(\aleph_2)$ and $φ(x, y)$ is a $Σ_0$ formula with the property that for every ground model $M$ of $CH$ with $a\in M$ there is, in $M$, a suitably nice poset -- specifically, a poset $\mathbb{Q}\subseteq\mathcal{H}(κ)^M$ which is $ω_1$-linked and symmetrically proper -- adding some $b$ such that $φ(a, b)$. In particular, $\mathbb{P}$ forces Moore's Measuring principle, Baumgartner's Axiom for $\aleph_1$-dense sets of reals, Todorčević's Open Colouring Axiom for sets of size $\aleph_1$, the Abraham-Rubin-Shelah Open Colouring Axiom, and Todorčević's P-ideal Dichotomy for $\aleph_1$-generated ideals on $ω_1$, among other statements. Hence, all these statements are simultaneously compatible with a large continuum. Finally, we show that a further small variation of our construction yields a model satisfying, in addition to all the earlier conclusions, Martin's Maximum for posets of size $\aleph_1$.

math.LO↗

arXiv2025

On the number of cofinalities of cuts in ultraproducts of linear orders

Suppose $κ$ is a regular cardinal and $\bar a=\langle μ_i: i<κ\rangle$ is a non-decreasing sequence of regular cardinals. We study the set of possible cofinalities of cuts Pcut$(\bar a)=\{(λ_1, λ_2):$ for some ultrafilter $D$ on $κ$, $(λ_1, λ_2)$ is the cofinality of a cut of $\prod\limits_{i<κ} μ_i / D \}$.

math.LO↗

arXiv2024

Expressive Power of Infinitary Logic and Absolute co-Hopfianity

Recently, Paolini and Shelah have constructed absolutely Hopfian torsion-free abelian groups of any given size. In contrast, we show that this is not necessarily the case for absolutely co-Hopfian groups. We use the infinitary logic to show that there are no absolute co-Hopfian abelian groups above the first beautiful cardinal. An extension of this result to the category of modules over a commutative ring is given.

math.LO↗

arXiv2024

Adding Abraham clubs and $α$-properness

For every indecomposable ordinal $α< ω_1$, we introduce a variant of Abraham forcing for adding a club in $ω_1$, which is $<α$-proper but not $α$-proper.

math.LO↗

arXiv2024

On groups well represented as automorphism groups of groups

Assuming Gödel's axiom of constructibility $\bold V=\bold L,$ we present a characterization of those groups $L$ for which there exist arbitrarily large groups $H$ such that $aut(H) \cong L$. In particular, we show that it suffices to have one such group $H$ such that the size of its center is bigger than $ 2^{|L |+\aleph_0}$.

math.GR↗

arXiv2024

On a Question of Hamkins and Löwe on the modal logic of collapse forcing

Hamkins and Löwe asked whether there can be a model $N$ of set theory with the property that $N\equiv N[g]$ whenever $g$ is a generic collapse of a cardinal of $N$ onto $ω$. We give equiconsistency results for two weaker versions of this property. We also include a proof of Woodin's result that the consistency of the full Hamkins-Löwe property follows from that of a Woodin cardinal with an inaccessible above.

math.LO↗

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.

math.LO↗