Adding a lot of random reals by adding a few
We study pairs $(V, V_1)$ of models of $ZFC$ such that adding $κ$-many random reals over $V_1$ adds $λ$-many random reals over $V$, for some $λ> κ.$
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arXiv subjects
Mohammad Golshani
Publications and source records attributed to Mohammad Golshani.
The tree property on a countable segment of successors of singular cardinals
Starting from the existence of many supercompact cardinals, we construct a model of ZFC in which the tree property holds at a countable segment of successor of singular cardinals.
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On a question of Silver about gap-two cardinal transfer principles
Assuming the existence of a Mahlo cardinal, we produce a generic extension of Gödel's constructible universe $L$, in which the transfer principles $(\aleph_2, \aleph_0) \to (\aleph_3, \aleph_1)$ and $(\aleph_3, \aleph_1) \to (\aleph_2, \aleph_0)$ fail simultaneously. The result answers a question of Silver from 1971. We also extend our result to higher gaps.
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Adding many random reals may add many Cohen reals
Let $κ$ be an infinite cardinal. Then, forcing with $\mathbb{R}(κ)$$\times$$\mathbb{R}(κ)$ adds a generic filter for $\mathbb{C}(κ);$ where $\mathbb{R}(κ)$ and $\mathbb{C}(κ)$ are the forcing notions for adding $κ$-many random reals and adding $κ$-many Cohen reals respectively.
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Notes on countably generated complete Boolean algebras
We give a necessary and sufficient condition for an atomless Boolean algebra to be countably generated, and use it to give new proofs of some some know facts due to Gaifman-Hales and Solovay and also due to Jech, Kunen and Magidor. We also show that Jensen's coding theorem can be used to provide cardinal preserving countably generated complete Boolean algebras of arbitrary large cardinality. This answers a question of Jech [5] from 1976.
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Tree property at successor of a singular limit of measurable cardinals
Assume $λ$ is a singular limit of $η$ supercompact cardinals, where $η\leq λ$ is a limit ordinal. We present two forcing methods for making $λ^+$ the successor of the limit of the first $η$ measurable cardinals while the tree property holding at $λ^+.$ The first method is then used to get, from the same assumptions, tree property at $\aleph_{η^2+1}$ with the failure of $SCH$ at $\aleph_{η^2}$. This extends results of Neeman and Sinapova. The second method is also used to get tree property at successor of an arbitrary singular cardinal, which extends some results of Magidor-Shelah, Neeman and Sinapova.
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An Easton like theorem in the presence of Shelah Cardinals
We show that Shelah cardinals are preserved under the canonical $GCH$ forcing notion. We also show that if $GCH$ holds and $F:REG\rightarrow CARD$ is an Easton function which satisfies some weak properties, then there exists a cofinality preserving generic extension of the universe which preserves Shelah cardinals and satisfies $\forall κ\in REG,~ 2^κ=F(κ)$. This gives a partial answer to a question asked by Cody [1] and independently by Honzik [5]. We also prove an indestructibility result for Shelah cardinals.
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On cuts in ultraproducts of linear orders I
For an ultrafilter $D$ on a cardinal $κ,$ we wonder for which pair $(θ_1, θ_2)$ of regular cardinals, we have: for any $(θ_1+θ_2)^+-$saturated dense linear order $J, J^κ/ D$ has a cut of cofinality $(θ_1, θ_2).$ We deal mainly with the case $θ_1, θ_2 > 2^κ.$
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All uncountable regular cardinals can be inaccessible in HOD
Assuming the existence of a supercompact cardinal and an inaccessible above it, we construct a model of ZFC, in which all uncountable regular cardinals are inaccessible in HOD.
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On Foreman's maximality principle
In this paper we consider the Foreman's maximality principle, which says that any non-trivial forcing notion either adds a new real or collapses some cardinals. We prove the consistency of some of its consequences. We prove that it is consistent that every $c.c.c.$ forcing adds a real and that for every uncountable regular cardinal $κ$, every $κ$-closed forcing of size $2^{<κ}$ collapses some cardinals.
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Power set at $\aleph_ω$: On a theorem of Woodin
We give Woodin's original proof that if there exists a $(κ+2)-$strong cardinal $κ,$ then there is a generic extension of the universe in which $κ=\aleph_ω,$ $GCH$ holds below $\aleph_ω$ and $2^{\aleph_ω}=\aleph_{ω+2}.$
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A Groszek-Laver pair of undistinguishable $E_0$ classes
A generic extension $L[x,y]$ of $L$ by reals $x,y$ is defined, in which the union of $\mathsf E_0$-classes of $x$ and $y$ is a $Π^1_2$ set, but neither of these two $\mathsf E_0$-classes is separately ordinal-definable.
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Collapsing the cardinals of $HOD$
Assuming that $GCH$ holds and $κ$ is $κ^{+3}$-supercompact, we construct a generic extension $W$ of $V$ in which $κ$ remains strongly inaccessible and $(α^+)^{HOD} < α^+$ for every infinite cardinal $α< κ$. In particular the rank-initial segment $W_κ$ is a model of ZFC in which $(α^+)^{HOD} < α^+$ for every infinite cardinal $α$.
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HOD, V and the GCH
Starting from large cardinals we construct a model of $ZFC$ in which the $GCH$ fails everywhere, but such that $GCH$ holds in its $HOD$. The result answers a question of Sy Friedman. Also, relative to the existence of large cardinals, we produce a model of $ZFC+GCH$ such that $GCH$ fails everywhere in its $HOD$.
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Almost Souslin Kurepa trees
We show that the existence of an almost Souslin Kurepa tree is consistent with $ZFC$. We also prove their existence in $L$. These results answer two questions from Zakrzewski.
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More on almost Souslin Kurepa trees
It is consistent that there exists a Souslin tree $T$ such that after forcing with it, $T$ becomes an almost Souslin Kurepa tree. This answers a question of Zakrzewski.
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Independence of higher Kurepa hypotheses
We study the Generalized Kurepa Hypothesis introduced by Chang. We show that relative to the existence of an inaccessible cardinal the Gap-$n$-Kurepa hypothesis does not follow from the Gap-$m$-Kurepa hypothesis for $m$ different from $n$. The use of an inaccessible is necessary for this result.
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