arXiv subjects
Publications and source records attributed to Rahman Mohammadpour.
In his Mostowski lecture in Wroc{\l}aw in 2024, Stevo Todor\v{c}evi\'c asked whether it is consistent that Rado's Conjecture holds at two successive cardinals. We show that it is consistent that Rado's Conjecture holds at all regular cardinals.
We show that the consistency of $\mathrm{ZF} + \mathrm{AD}_{\mathbb{R}} + ``\Theta$ is measurable$"$ implies the consistency of $\mathrm{ZF} +``\Theta$ is the least strongly regular cardinal and the least measurable cardinal$"$ + $``$all uncountable cardinals below $\Theta$ are of countable cofinality$"$.
I show that it is consistent relative to the consistency of a Mahlo cardinal that Martin's axiom holds at $\omega_2$, but the weak Kurepa Hypothesis fails. This answers a question posed by Honzik, Lambie-Hanson and Stejskalov\'a. The consistency result is obtained by constructing a model where the weak Kurepa Hypothesis fails in any c.c.c. forcing extension.
We show that the existence of a well-known type of ideals on a regular cardinal $λ$ implies a compactness property concerning the specialisability of a tree of height $λ$ with no cofinal branches. We also use Neeman's method of side conditions to show that the existence of such ideals is consistent with stationarily many appropriate guessing models. These objects suffice to extend the main theorem of \cite{mhpr_spe}: one can generically specialise any branchless tree of height $κ^{++}$ with a ${<}κ$-closed, $κ^{+}$-proper, and $κ^{++}$-preserving forcing, which has the $κ^+$-approximation property.
In \cite{MV} we defined and proved the consistency of the principle ${\rm GM}^+(\omega_3,\omega_1)$ which implies that many consequences of strong forcing axioms hold simultaneously at $\omega_2$ and $\omega_3$. In this paper we formulate a strengthening of ${\rm GM}^+(\omega_3,\omega_1)$ that we call ${\rm SGM}^+(\omega_3,\omega_1)$. We also prove, modulo the consistency of two supercompact cardinals, that ${\rm SGM}^+(\omega_3,\omega_1)$ is consistent with ZFC. In addition to all the consequences of ${\rm GM}^+(\omega_3,\omega_1)$, the principle ${\rm SGM}^+(\omega_3,\omega_1)$, together with some mild cardinal arithmetic assumptions that hold in our model, implies that any forcing that adds a new subset of $\omega_2$ either adds a real or collapses some cardinal. This gives a partial answer to a question of Abraham \cite{AvrahamPhD} and extends a previous result of Todor\v{c}evi\'{c} \cite{Todorcevic82} in this direction.
The compactness phenomenon is one of the featured aspects of structuralism in mathematics. In simple and broad words, a compactness property holds in a structure if a related property is satisfied by sufficiently many substructures of that structure. With this phenomenon and its twin sibling "reflection", modern set theory has settled many mathematical statements left undecided by the conventionally accepted formalism of mathematics, $\rm ZFC$. A broad research program investigates whether a notion of compactness can universe-widely emerge without running into contradictions. These notes are a survey about guessing models whose existence provides intriguing compactness phenomena. Most of the results in the manuscript are well-known. We shall reformulate, generalise and expand some of them. We also present some known applications of guessing models and state some open problems.
Assuming $\rm PFA$, we shall use internally club $ω_1$-guessing models as side conditions to show that for every tree $T$ of height $ω_2$ without cofinal branches, there is a proper and $\aleph_2$-preserving forcing notion with finite conditions which specialises $T$. Moreover, the forcing has the $ω_1$-approximation property.
In this short note, we shall prove some observations regarding the connection between indestructible $ω_1$-guessing models and the $ω_1$-approximation property of forcing notions.
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 \[ \prod_{n<ω}Add(κ_n,1)/fin \simeq Add(κ^+, 1). \]
We introduce the forcing property "almost strong properness" which sits between properness and strong properness. As an application, we introduce a simple forcing with finite conditions to force $\rm MRP$.
Starting with two supercompact cardinals we produce a generic extension of the universe in which a principle that we call ${\rm GM}^+(ω_3,ω_1)$ holds. This principle implies ${\rm ISP}(ω_2)$ and ${\rm ISP}(ω_3)$, and hence the tree property at $ω_2$ and $ω_3$, the Singular Cardinal Hypothesis, and the failure of the weak square principle $\square(ω_2,λ)$, for all regular $λ\geq ω_2$. In addition, it implies that the restriction of the approachability ideal $I[ω_2]$ to the set of ordinals of cofinality $ω_1$ is the non stationary ideal on this set. The consistency of this last statement was previously shown by Mitchell.
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 $κ^{++}$.
We present new, streamlined proofs of certain maximality principles studied by Hamkins and Woodin. Moreover, we formulate an intermediate maximality principle, which is shown here to be equiconsistent with the existence of a weakly compact cardinal $κ$ such that $V_κ\prec V$.