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Pcf without choice Sh835
We mainly investigate models of set theory with restricted choice, e.g., ZF + DC + the family of countable subsets of \(\lambda \) ...
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Article
Open AccessMany forcing axioms for all regular uncountable cardinals
A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and having a strong forcing axiom of higher order than usual. Instead o...
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Article
Shearing in some simple rank one theories
Dividing asks about inconsistency along indiscernible sequences. In order to study the finer structure of simple theories without much dividing, the authors recently introduced shearing, which essentially asks...
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On the non-existence of \(\kappa \) -mad families
Starting from a model with a Laver-indestructible supercompact cardinal \(\kappa \) κ ...
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Open AccessOn the existence of uncountable Hopfian and co-Hopfian abelian groups
We deal with the problem of existence of uncountable co-Hopfian abelian groups and (absolute) Hopfian abelian groups. Firstly, we prove that there are no co-Hopfian reduced abelian groups G of size < p with infin...
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Article
Corrected iteration
For \(\lambda \) λ inaccessible, we may consider
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Open AccessAutomorphisms and strongly invariant relations
We investigate characterizations of the Galois connection \({{\,\textrm{Aut}\,}}\) ...
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The Keisler–Shelah isomorphism theorem and the continuum hypothesis II
We continue the investigation started in Golshani (2021) about the relation between the Keilser–Shelah isomorphism theorem and the continuum hypothesis. In particular, we show it is consistent that the continu...
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Universality: new criterion for non-existence
We find new “reasons” for a class of models for not having a universal model in a cardinal \(\lambda \) λ ...
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Open AccessPositive logics
Lindström’s Theorem characterizes first order logic as the maximal logic satisfying the Compactness Theorem and the Downward Löwenheim-Skolem Theorem. If we do not assume that logics are closed under negation,...
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Hanf numbers for extendibility and related phenomena
This paper contains portions of Baldwin’s talk at the Set Theory and Model Theory Conference (Institute for Research in Fundamental Sciences, Tehran, October 2015) and a detailed proof that in a suitable exten...
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The independence of \(\mathsf {GCH}\) and a combinatorial principle related to Banach–Mazur games
It was proved recently that Telgársky’s conjecture, which concerns partial information strategies in the Banach–Mazur game, fails in models of ...
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Isomorphic limit ultrapowers for infinitary logic
The logic \({\mathbb{L}}_\theta ^1\) L θ 1 ...
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Open AccessPreservation of splitting families and cardinal characteristics of the continuum
We show how to construct, via forcing, splitting families that are preserved by a certain type of finite support iterations. As an application, we construct a model where 15 classical characteristics of the co...
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The cofinality of the symmetric group and the cofinality of ultrapowers
We prove that \(\mathfrak{mcf}\) m c f ...
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On some variants of the club principle
We study some asymptotic variants of the club principle. Along the way, we construct some forcings and use them to separate several of these principles.
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Open AccessA version of \(\kappa \) -Miller forcing
We consider a version of \(\kappa \) κ -Miller forcing on an uncountable cardinal \(\kappa \) κ . We show that under \(2^{<\kappa } = \kappa \) 2 < κ = κ this forcing collapses \(2^\kappa \) 2
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Retraction Note of “Baire property and axiom of choice”
I hereby retract the paper [1] as the proof has an irreparable gap.
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Quite free complicated abelian groups, pcf and black boxes
We would like to build Abelian groups (or R-modules) which on the one hand are quite free, say ℵω+1-free, and on the other hand are complicated in a suitable sense. We choose as our test problem one having no non...
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Article
No universal in singular
An old question is whether there is a countable complete first order theory T such that T has a universal model of cardinality \(\lambda> {\aleph _0} {\underline{iff}}\, \lambda = 2^{< \lambda } > {\aleph _0} \)
